diff --git a/include/geom/cell_hex.h b/include/geom/cell_hex.h
index 33d5933ad80..1f77951794c 100644
--- a/include/geom/cell_hex.h
+++ b/include/geom/cell_hex.h
@@ -125,11 +125,18 @@ class Hex : public Cell
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
   /**
    * \returns \p Hex8::side_nodes_map[side][side_node] after doing some range checking.
    */
diff --git a/include/geom/cell_hex27.h b/include/geom/cell_hex27.h
index 4781b98e7be..8a50372593a 100644
--- a/include/geom/cell_hex27.h
+++ b/include/geom/cell_hex27.h
@@ -151,8 +151,7 @@ class Hex27 final : public Hex
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Hex27 since we can
    * use the center node of each face to provide a perfect (unique)
diff --git a/include/geom/cell_inf_hex.h b/include/geom/cell_inf_hex.h
index 12903135269..423f3a416dc 100644
--- a/include/geom/cell_inf_hex.h
+++ b/include/geom/cell_inf_hex.h
@@ -151,11 +151,18 @@ class InfHex : public InfCell
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
   /**
    * \returns \p InfHex8::side_nodes_map[side][side_node] after doing some range checking.
    */
diff --git a/include/geom/cell_inf_hex18.h b/include/geom/cell_inf_hex18.h
index 9400e9091e8..737ce73531e 100644
--- a/include/geom/cell_inf_hex18.h
+++ b/include/geom/cell_inf_hex18.h
@@ -160,8 +160,7 @@ class InfHex18 final : public InfHex
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p InfHex18 since we can
    * use the center node of the bottom face to provide a perfect (unique)
diff --git a/include/geom/cell_inf_prism.h b/include/geom/cell_inf_prism.h
index d67809fe355..5b953eb1f9f 100644
--- a/include/geom/cell_inf_prism.h
+++ b/include/geom/cell_inf_prism.h
@@ -143,11 +143,18 @@ class InfPrism : public InfCell
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
   /**
    * \returns InfPrism6::side_nodes_map[side][side_node] after doing some range checking.
    */
diff --git a/include/geom/cell_prism.h b/include/geom/cell_prism.h
index 09f3ad2e8ad..52d2bed1b1b 100644
--- a/include/geom/cell_prism.h
+++ b/include/geom/cell_prism.h
@@ -117,11 +117,18 @@ class Prism : public Cell
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
   /**
    * \returns \p Prism6::side_nodes_map[side][side_node] after doing some range checking.
    */
diff --git a/include/geom/cell_prism18.h b/include/geom/cell_prism18.h
index 532d5264a47..722c00325e9 100644
--- a/include/geom/cell_prism18.h
+++ b/include/geom/cell_prism18.h
@@ -156,8 +156,7 @@ class Prism18 final : public Prism
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Prism18 since we can
    * use the center node of each quad face to provide a perfect (unique)
diff --git a/include/geom/cell_prism20.h b/include/geom/cell_prism20.h
index f1a954ee284..e6c666e82b9 100644
--- a/include/geom/cell_prism20.h
+++ b/include/geom/cell_prism20.h
@@ -160,8 +160,7 @@ class Prism20 final : public Prism
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Prism20 since we can
    * use the center node of each quad face to provide a perfect (unique)
diff --git a/include/geom/cell_prism21.h b/include/geom/cell_prism21.h
index f2c3749792b..742d2795972 100644
--- a/include/geom/cell_prism21.h
+++ b/include/geom/cell_prism21.h
@@ -163,8 +163,7 @@ class Prism21 final : public Prism
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Prism21 since we can
    * use the center node of each quad face to provide a perfect (unique)
diff --git a/include/geom/cell_pyramid.h b/include/geom/cell_pyramid.h
index 25898ec3160..5a70b5fd43d 100644
--- a/include/geom/cell_pyramid.h
+++ b/include/geom/cell_pyramid.h
@@ -120,14 +120,21 @@ class Pyramid : public Cell
    */
   using Elem::key;
 
-  /**
+   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
   /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
+ /**
    * \returns \p Pyramid5::side_nodes_map[side][side_node] after doing some range checking.
    */
   virtual unsigned int local_side_node(unsigned int side,
diff --git a/include/geom/cell_pyramid14.h b/include/geom/cell_pyramid14.h
index 30d8cf3cf11..348959299a5 100644
--- a/include/geom/cell_pyramid14.h
+++ b/include/geom/cell_pyramid14.h
@@ -158,8 +158,7 @@ class Pyramid14 final : public Pyramid
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Pyramid14 since we can
    * use the center node of the base face to provide a perfect (unique)
diff --git a/include/geom/cell_pyramid18.h b/include/geom/cell_pyramid18.h
index d1781a1ab38..f28fc71f034 100644
--- a/include/geom/cell_pyramid18.h
+++ b/include/geom/cell_pyramid18.h
@@ -164,8 +164,7 @@ class Pyramid18 final : public Pyramid
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Pyramid18 since we can
    * use the center node of the base face to provide a perfect (unique)
diff --git a/include/geom/cell_tet.h b/include/geom/cell_tet.h
index 2916d98808a..5cae533edd0 100644
--- a/include/geom/cell_tet.h
+++ b/include/geom/cell_tet.h
@@ -106,12 +106,19 @@ class Tet : public Cell
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
   /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
+ /**
    * \returns \p Tet4::side_nodes_map[side][side_node] after doing some range checking.
    */
   virtual unsigned int local_side_node(unsigned int side,
diff --git a/include/geom/edge.h b/include/geom/edge.h
index 339d22e3d12..d8890e83904 100644
--- a/include/geom/edge.h
+++ b/include/geom/edge.h
@@ -127,12 +127,20 @@ class Edge : public Elem
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override final
   { return this->compute_key(this->node_id(s)); }
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override final
+  { return this->compute_key(this->node_id(s)); }
+
   /**
    * \returns \p side after doing some range checking. \p side_node is ignored.
    */
diff --git a/include/geom/elem.h b/include/geom/elem.h
index 1eac5a60d0b..2ea1fe9ae16 100644
--- a/include/geom/elem.h
+++ b/include/geom/elem.h
@@ -249,11 +249,18 @@ class Elem : public ReferenceCountedObject<Elem>,
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const = 0;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const = 0;
+
   /**
    * \returns An id associated with the global node ids of this
    * element.  The id is not necessarily unique, but should be
diff --git a/include/geom/face_inf_quad.h b/include/geom/face_inf_quad.h
index 55035035304..8dd4ec33dcd 100644
--- a/include/geom/face_inf_quad.h
+++ b/include/geom/face_inf_quad.h
@@ -157,11 +157,18 @@ class InfQuad : public Elem
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
+  /**
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
+   */
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
+
   /**
    * \returns \p InfQuad4::side_nodes_map[side][side_node] after doing some range checking.
    */
diff --git a/include/geom/face_inf_quad6.h b/include/geom/face_inf_quad6.h
index a9ea06818e3..e0cded568ad 100644
--- a/include/geom/face_inf_quad6.h
+++ b/include/geom/face_inf_quad6.h
@@ -113,8 +113,7 @@ class InfQuad6 : public InfQuad
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We override this function to return a key for the Edge3 side
    * which is consistent which Edge3::key().
diff --git a/include/geom/face_quad.h b/include/geom/face_quad.h
index 6eab8c5bdc0..4f738934b12 100644
--- a/include/geom/face_quad.h
+++ b/include/geom/face_quad.h
@@ -131,19 +131,26 @@ class Quad : public Face
    */
   using Elem::key;
 
+  /**
+   * \returns An id associated with the global node ids of this
+   * element.  The id is not necessarily unique, but should be
+   * close.
+   */
+  virtual dof_id_type key () const override;
+
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
   /**
-   * \returns An id associated with the global node ids of this
-   * element.  The id is not necessarily unique, but should be
-   * close.
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
    */
-  virtual dof_id_type key () const override;
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
 
   /**
    * \returns \p Quad4::side_nodes_map[side][side_node] after doing some range checking.
diff --git a/include/geom/face_quad8.h b/include/geom/face_quad8.h
index fc7f28ed869..9e7ff4773e0 100644
--- a/include/geom/face_quad8.h
+++ b/include/geom/face_quad8.h
@@ -132,8 +132,7 @@ class Quad8 : public Quad
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Quad8 since we can
    * use the center node of each edge to provide a perfect (unique)
diff --git a/include/geom/face_quad9.h b/include/geom/face_quad9.h
index 358b50767b4..71135a346e1 100644
--- a/include/geom/face_quad9.h
+++ b/include/geom/face_quad9.h
@@ -132,8 +132,7 @@ class Quad9 : public Quad
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
    * We reimplement this method here for the \p Quad9 since we can
    * use the center node of each edge to provide a perfect (unique)
diff --git a/include/geom/face_tri.h b/include/geom/face_tri.h
index defd07aeb23..355576c963c 100644
--- a/include/geom/face_tri.h
+++ b/include/geom/face_tri.h
@@ -118,19 +118,26 @@ class Tri : public Face
    */
   using Elem::key;
 
+  /**
+   * \returns An id associated with the global node ids of this
+   * element.  The id is not necessarily unique, but should be
+   * close.
+   */
+  virtual dof_id_type key () const override final;
+
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    */
   virtual dof_id_type key (const unsigned int s) const override;
 
   /**
-   * \returns An id associated with the global node ids of this
-   * element.  The id is not necessarily unique, but should be
-   * close.
+   * \returns An id associated with the \p s side of this element, as
+   * defined solely by element vertices.  The id is not necessarily
+   * unique, but should be close.  This is particularly useful in the
+   * \p MeshBase::find_neighbors() routine.
    */
-  virtual dof_id_type key () const override final;
+  virtual dof_id_type low_order_key (const unsigned int s) const override;
 
   /**
    * \returns \p Tri3::side_nodes_map[side][side_node] after doing some range checking.
diff --git a/include/geom/face_tri6.h b/include/geom/face_tri6.h
index 40b93b9f81f..87894130c3b 100644
--- a/include/geom/face_tri6.h
+++ b/include/geom/face_tri6.h
@@ -142,10 +142,9 @@ class Tri6 : public Tri
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
-   * We reimplement this method here for the \p Quad8 since we can
+   * We reimplement this method here for the \p Tri6 since we can
    * use the center node of each edge to provide a perfect (unique)
    * key.
    */
diff --git a/include/geom/face_tri7.h b/include/geom/face_tri7.h
index d4c6151d97f..345f775ac90 100644
--- a/include/geom/face_tri7.h
+++ b/include/geom/face_tri7.h
@@ -147,10 +147,9 @@ class Tri7 : public Tri
 
   /**
    * \returns An id associated with the \p s side of this element.
-   * The id is not necessarily unique, but should be close.  This is
-   * particularly useful in the \p MeshBase::find_neighbors() routine.
+   * The id is not necessarily unique, but should be close.
    *
-   * We reimplement this method here for the \p Quad8 since we can
+   * We reimplement this method here for the \p Tri7 since we can
    * use the center node of each edge to provide a perfect (unique)
    * key.
    */
diff --git a/include/geom/node_elem.h b/include/geom/node_elem.h
index 04ded413581..8ce404dc270 100644
--- a/include/geom/node_elem.h
+++ b/include/geom/node_elem.h
@@ -114,6 +114,13 @@ class NodeElem : public Elem
   virtual dof_id_type key (const unsigned int) const override
   { libmesh_error_msg("Calling NodeElem::key(side) does not make sense."); return 0; }
 
+  /**
+   * \returns An id associated with the \p s side of this element.
+   * This should never be important for NodeElems.
+   */
+  virtual dof_id_type low_order_key (const unsigned int) const override
+  { libmesh_error_msg("Calling NodeElem::low_order_key(side) does not make sense."); return 0; }
+
   /**
    * NodeElems don't have sides, so they can't have nodes on sides.
    */
diff --git a/include/geom/remote_elem.h b/include/geom/remote_elem.h
index 12fabd09611..68391765ad7 100644
--- a/include/geom/remote_elem.h
+++ b/include/geom/remote_elem.h
@@ -109,6 +109,9 @@ class RemoteElem : public Elem,
   virtual dof_id_type key (const unsigned int) const override
   { remote_elem_error("key"); return 0; }
 
+  virtual dof_id_type low_order_key (const unsigned int) const override
+  { remote_elem_error("low_order_key"); return 0; }
+
   virtual unsigned int local_side_node(unsigned int /*side*/,
                                        unsigned int /*side_node*/) const override
   { remote_elem_error("local_side_node"); return 0; }
diff --git a/src/geom/cell_hex.C b/src/geom/cell_hex.C
index e321a01e99a..414f41c9369 100644
--- a/src/geom/cell_hex.C
+++ b/src/geom/cell_hex.C
@@ -99,6 +99,18 @@ dof_id_type Hex::key (const unsigned int s) const
 
 
 
+dof_id_type Hex::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  return this->compute_key(this->node_id(Hex8::side_nodes_map[s][0]),
+                           this->node_id(Hex8::side_nodes_map[s][1]),
+                           this->node_id(Hex8::side_nodes_map[s][2]),
+                           this->node_id(Hex8::side_nodes_map[s][3]));
+}
+
+
+
 unsigned int Hex::local_side_node(unsigned int side,
                                   unsigned int side_node) const
 {
diff --git a/src/geom/cell_inf_hex.C b/src/geom/cell_inf_hex.C
index d785c3beb6d..d61435ce420 100644
--- a/src/geom/cell_inf_hex.C
+++ b/src/geom/cell_inf_hex.C
@@ -97,6 +97,20 @@ dof_id_type InfHex::key (const unsigned int s) const
 
 
 
+dof_id_type InfHex::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  // The order of the node ids does not matter, they are sorted by the
+  // compute_key() function.
+  return this->compute_key(this->node_id(InfHex8::side_nodes_map[s][0]),
+                           this->node_id(InfHex8::side_nodes_map[s][1]),
+                           this->node_id(InfHex8::side_nodes_map[s][2]),
+                           this->node_id(InfHex8::side_nodes_map[s][3]));
+}
+
+
+
 unsigned int InfHex::local_side_node(unsigned int side,
                                      unsigned int side_node) const
 {
diff --git a/src/geom/cell_inf_prism.C b/src/geom/cell_inf_prism.C
index df04263ce7e..bbc0118434a 100644
--- a/src/geom/cell_inf_prism.C
+++ b/src/geom/cell_inf_prism.C
@@ -92,6 +92,32 @@ dof_id_type InfPrism::key (const unsigned int s) const
 
 
 
+dof_id_type InfPrism::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  switch (s)
+    {
+    case 0: // the triangular face at z=-1, base face
+      return this->compute_key (this->node_id(InfPrism6::side_nodes_map[s][0]),
+                                this->node_id(InfPrism6::side_nodes_map[s][1]),
+                                this->node_id(InfPrism6::side_nodes_map[s][2]));
+
+    case 1: // the quad face at y=0
+    case 2: // the other quad face
+    case 3: // the quad face at x=0
+      return this->compute_key (this->node_id(InfPrism6::side_nodes_map[s][0]),
+                                this->node_id(InfPrism6::side_nodes_map[s][1]),
+                                this->node_id(InfPrism6::side_nodes_map[s][2]),
+                                this->node_id(InfPrism6::side_nodes_map[s][3]));
+
+    default:
+      libmesh_error_msg("Invalid side s = " << s);
+    }
+}
+
+
+
 unsigned int InfPrism::local_side_node(unsigned int side,
                                        unsigned int side_node) const
 {
diff --git a/src/geom/cell_prism.C b/src/geom/cell_prism.C
index 44123f8b579..b484834a5e2 100644
--- a/src/geom/cell_prism.C
+++ b/src/geom/cell_prism.C
@@ -97,6 +97,33 @@ dof_id_type Prism::key (const unsigned int s) const
 
 
 
+dof_id_type Prism::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  switch (s)
+    {
+    case 0: // the triangular face at z=0
+    case 4: // the triangular face at z=1
+      return this->compute_key (this->node_id(Prism6::side_nodes_map[s][0]),
+                                this->node_id(Prism6::side_nodes_map[s][1]),
+                                this->node_id(Prism6::side_nodes_map[s][2]));
+
+    case 1: // the quad face at y=0
+    case 2: // the other quad face
+    case 3: // the quad face at x=0
+      return this->compute_key (this->node_id(Prism6::side_nodes_map[s][0]),
+                                this->node_id(Prism6::side_nodes_map[s][1]),
+                                this->node_id(Prism6::side_nodes_map[s][2]),
+                                this->node_id(Prism6::side_nodes_map[s][3]));
+
+    default:
+      libmesh_error_msg("Invalid side " << s);
+    }
+}
+
+
+
 unsigned int Prism::local_side_node(unsigned int side,
                                     unsigned int side_node) const
 {
diff --git a/src/geom/cell_pyramid.C b/src/geom/cell_pyramid.C
index cf7e234a0e2..c34b3ffa4d8 100644
--- a/src/geom/cell_pyramid.C
+++ b/src/geom/cell_pyramid.C
@@ -91,6 +91,33 @@ dof_id_type Pyramid::key (const unsigned int s) const
 
 
 
+dof_id_type Pyramid::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  switch (s)
+    {
+    case 0: // triangular face 1
+    case 1: // triangular face 2
+    case 2: // triangular face 3
+    case 3: // triangular face 4
+      return this->compute_key (this->node_id(Pyramid5::side_nodes_map[s][0]),
+                                this->node_id(Pyramid5::side_nodes_map[s][1]),
+                                this->node_id(Pyramid5::side_nodes_map[s][2]));
+
+    case 4:  // the quad face at z=0
+      return this->compute_key (this->node_id(Pyramid5::side_nodes_map[s][0]),
+                                this->node_id(Pyramid5::side_nodes_map[s][1]),
+                                this->node_id(Pyramid5::side_nodes_map[s][2]),
+                                this->node_id(Pyramid5::side_nodes_map[s][3]));
+
+    default:
+      libmesh_error_msg("Invalid side s = " << s);
+    }
+}
+
+
+
 unsigned int Pyramid::local_side_node(unsigned int side,
                                       unsigned int side_node) const
 {
diff --git a/src/geom/cell_tet.C b/src/geom/cell_tet.C
index 390f43aec0c..079855f7d0b 100644
--- a/src/geom/cell_tet.C
+++ b/src/geom/cell_tet.C
@@ -69,6 +69,17 @@ dof_id_type Tet::key (const unsigned int s) const
 
 
 
+dof_id_type Tet::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  return this->compute_key(this->node_id(Tet4::side_nodes_map[s][0]),
+                           this->node_id(Tet4::side_nodes_map[s][1]),
+                           this->node_id(Tet4::side_nodes_map[s][2]));
+}
+
+
+
 unsigned int Tet::local_side_node(unsigned int side,
                                   unsigned int side_node) const
 {
diff --git a/src/geom/face_inf_quad.C b/src/geom/face_inf_quad.C
index 5770080b9af..76f67c71809 100644
--- a/src/geom/face_inf_quad.C
+++ b/src/geom/face_inf_quad.C
@@ -61,6 +61,18 @@ dof_id_type InfQuad::key (const unsigned int s) const
 
 
 
+dof_id_type InfQuad::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  // The order of the node ids does not matter, they are sorted by the
+  // compute_key() function.
+  return this->compute_key(this->node_id(InfQuad4::side_nodes_map[s][0]),
+                           this->node_id(InfQuad4::side_nodes_map[s][1]));
+}
+
+
+
 unsigned int InfQuad::local_side_node(unsigned int side,
                                       unsigned int side_node) const
 {
diff --git a/src/geom/face_quad.C b/src/geom/face_quad.C
index cb9a8ad0643..b7178f21845 100644
--- a/src/geom/face_quad.C
+++ b/src/geom/face_quad.C
@@ -62,6 +62,16 @@ dof_id_type Quad::key (const unsigned int s) const
 
 
 
+dof_id_type Quad::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  return this->compute_key(this->node_id(Quad4::side_nodes_map[s][0]),
+                           this->node_id(Quad4::side_nodes_map[s][1]));
+}
+
+
+
 unsigned int Quad::local_side_node(unsigned int side,
                                    unsigned int side_node) const
 {
diff --git a/src/geom/face_tri.C b/src/geom/face_tri.C
index 9b628b26109..ff59c86d836 100644
--- a/src/geom/face_tri.C
+++ b/src/geom/face_tri.C
@@ -61,6 +61,16 @@ dof_id_type Tri::key (const unsigned int s) const
 
 
 
+dof_id_type Tri::low_order_key (const unsigned int s) const
+{
+  libmesh_assert_less (s, this->n_sides());
+
+  return this->compute_key(this->node_id(Tri3::side_nodes_map[s][0]),
+                           this->node_id(Tri3::side_nodes_map[s][1]));
+}
+
+
+
 unsigned int Tri::local_side_node(unsigned int side,
                                   unsigned int side_node) const
 {
diff --git a/src/mesh/boundary_info.C b/src/mesh/boundary_info.C
index 8aa1e8595bb..b3d35ffcd79 100644
--- a/src/mesh/boundary_info.C
+++ b/src/mesh/boundary_info.C
@@ -3228,19 +3228,29 @@ void BoundaryInfo::clear_stitched_boundary_side_ids (const boundary_id_type side
     auto pred_result = predicate(*it, _boundary_side_id);
     if (pred_result.first)
     {
-      // First erase associated nodeset information
+      // First erase associated nodeset information.  Do it from both
+      // sides, so we get any higher-order nodes if we're looking at
+      // them from a lower-order side, and so we only remove the two
+      // boundary ids used for stitching.
       if (clear_nodeset_data)
       {
         const Elem & elem = *it->first;
+        const Elem & neigh = *pred_result.second->first;
         const auto elem_side = it->second.first;
-        const auto local_node_nums = elem.nodes_on_side(elem_side);
-        for (const auto local_node_num : local_node_nums)
-          // This will remove all boundary info associated with this node, e.g. nodeset info
-          // for both sideset_id and other_sideset_id which is what we want
-          this->remove(elem.node_ptr(local_node_num));
+        const boundary_id_type neigh_side = pred_result.second->second.first;
+        const auto elem_bcid = it->second.second;
+        const boundary_id_type neigh_bcid = pred_result.second->second.second;
+
+        for (const auto local_node_num : elem.nodes_on_side(elem_side))
+          this->remove_node(elem.node_ptr(local_node_num), elem_bcid);
+
+        for (const auto local_node_num : neigh.nodes_on_side(neigh_side))
+          this->remove_node(neigh.node_ptr(local_node_num), neigh_bcid);
       }
 
-      // Now erase the sideset information
+      // Now erase the sideset information for our element and its
+      // neighbor, together.  This is safe since a multimap doesn't
+      // invalidate iterators.
       _boundary_side_id.erase(pred_result.second);
       it = _boundary_side_id.erase(it);
     }
diff --git a/src/mesh/unstructured_mesh.C b/src/mesh/unstructured_mesh.C
index d7226d728a1..d9d621dcb45 100644
--- a/src/mesh/unstructured_mesh.C
+++ b/src/mesh/unstructured_mesh.C
@@ -836,8 +836,10 @@ void UnstructuredMesh::find_neighbors (const bool reset_remote_elements,
             if (element->neighbor_ptr(ms) == nullptr ||
                 element->neighbor_ptr(ms) == remote_elem)
               {
-                // Get the key for the side of this element
-                const dof_id_type key = element->key(ms);
+                // Get the key for the side of this element.  Use the
+                // low_order_key so we can find neighbors in
+                // mixed-order meshes if necessary.
+                const dof_id_type key = element->low_order_key(ms);
 
                 // Look for elements that have an identical side key
                 auto bounds = side_to_elem_map.equal_range(key);
@@ -1863,7 +1865,7 @@ UnstructuredMesh::stitching_helper (const MeshBase * other_mesh,
                           // This side is on the boundary, add its information to side_to_elem
                           if (skip_find_neighbors && (i==0))
                             {
-                              key_type key = el->key(side_id);
+                              key_type key = el->low_order_key(side_id);
                               val_type val;
                               val.first = el;
                               val.second = cast_int<unsigned char>(side_id);
@@ -2407,7 +2409,7 @@ UnstructuredMesh::stitching_helper (const MeshBase * other_mesh,
                     {
                       if (el->neighbor_ptr(s) == nullptr)
                         {
-                          key_type key = el->key(s);
+                          key_type key = el->low_order_key(s);
                           auto bounds = side_to_elem_map.equal_range(key);
 
                           if (bounds.first != bounds.second)
diff --git a/tests/Makefile.am b/tests/Makefile.am
index f4ca10fe134..fb57c3520ff 100644
--- a/tests/Makefile.am
+++ b/tests/Makefile.am
@@ -76,6 +76,7 @@ unit_tests_sources = \
   mesh/mesh_stitch.C \
   mesh/mesh_triangulation.C \
   mesh/mixed_dim_mesh_test.C \
+  mesh/mixed_order_test.C \
   mesh/nodal_neighbors.C \
   mesh/libmesh_poly2tri.C \
   mesh/slit_mesh_test.C \
diff --git a/tests/Makefile.in b/tests/Makefile.in
index 0a9a2646ce9..6d4f69ca97c 100644
--- a/tests/Makefile.in
+++ b/tests/Makefile.in
@@ -220,9 +220,9 @@ am__unit_tests_dbg_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
 	mesh/write_nodeset_data.C mesh/write_edgeset_data.C \
@@ -327,6 +327,7 @@ am__objects_3 = unit_tests_dbg-driver.$(OBJEXT) \
 	mesh/unit_tests_dbg-mesh_stitch.$(OBJEXT) \
 	mesh/unit_tests_dbg-mesh_triangulation.$(OBJEXT) \
 	mesh/unit_tests_dbg-mixed_dim_mesh_test.$(OBJEXT) \
+	mesh/unit_tests_dbg-mixed_order_test.$(OBJEXT) \
 	mesh/unit_tests_dbg-nodal_neighbors.$(OBJEXT) \
 	mesh/unit_tests_dbg-libmesh_poly2tri.$(OBJEXT) \
 	mesh/unit_tests_dbg-slit_mesh_test.$(OBJEXT) \
@@ -419,9 +420,9 @@ am__unit_tests_devel_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
 	mesh/write_nodeset_data.C mesh/write_edgeset_data.C \
@@ -524,6 +525,7 @@ am__objects_5 = unit_tests_devel-driver.$(OBJEXT) \
 	mesh/unit_tests_devel-mesh_stitch.$(OBJEXT) \
 	mesh/unit_tests_devel-mesh_triangulation.$(OBJEXT) \
 	mesh/unit_tests_devel-mixed_dim_mesh_test.$(OBJEXT) \
+	mesh/unit_tests_devel-mixed_order_test.$(OBJEXT) \
 	mesh/unit_tests_devel-nodal_neighbors.$(OBJEXT) \
 	mesh/unit_tests_devel-libmesh_poly2tri.$(OBJEXT) \
 	mesh/unit_tests_devel-slit_mesh_test.$(OBJEXT) \
@@ -612,9 +614,9 @@ am__unit_tests_oprof_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
 	mesh/write_nodeset_data.C mesh/write_edgeset_data.C \
@@ -717,6 +719,7 @@ am__objects_7 = unit_tests_oprof-driver.$(OBJEXT) \
 	mesh/unit_tests_oprof-mesh_stitch.$(OBJEXT) \
 	mesh/unit_tests_oprof-mesh_triangulation.$(OBJEXT) \
 	mesh/unit_tests_oprof-mixed_dim_mesh_test.$(OBJEXT) \
+	mesh/unit_tests_oprof-mixed_order_test.$(OBJEXT) \
 	mesh/unit_tests_oprof-nodal_neighbors.$(OBJEXT) \
 	mesh/unit_tests_oprof-libmesh_poly2tri.$(OBJEXT) \
 	mesh/unit_tests_oprof-slit_mesh_test.$(OBJEXT) \
@@ -805,9 +808,9 @@ am__unit_tests_opt_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
 	mesh/write_nodeset_data.C mesh/write_edgeset_data.C \
@@ -910,6 +913,7 @@ am__objects_9 = unit_tests_opt-driver.$(OBJEXT) \
 	mesh/unit_tests_opt-mesh_stitch.$(OBJEXT) \
 	mesh/unit_tests_opt-mesh_triangulation.$(OBJEXT) \
 	mesh/unit_tests_opt-mixed_dim_mesh_test.$(OBJEXT) \
+	mesh/unit_tests_opt-mixed_order_test.$(OBJEXT) \
 	mesh/unit_tests_opt-nodal_neighbors.$(OBJEXT) \
 	mesh/unit_tests_opt-libmesh_poly2tri.$(OBJEXT) \
 	mesh/unit_tests_opt-slit_mesh_test.$(OBJEXT) \
@@ -998,9 +1002,9 @@ am__unit_tests_prof_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
 	mesh/write_nodeset_data.C mesh/write_edgeset_data.C \
@@ -1103,6 +1107,7 @@ am__objects_11 = unit_tests_prof-driver.$(OBJEXT) \
 	mesh/unit_tests_prof-mesh_stitch.$(OBJEXT) \
 	mesh/unit_tests_prof-mesh_triangulation.$(OBJEXT) \
 	mesh/unit_tests_prof-mixed_dim_mesh_test.$(OBJEXT) \
+	mesh/unit_tests_prof-mixed_order_test.$(OBJEXT) \
 	mesh/unit_tests_prof-nodal_neighbors.$(OBJEXT) \
 	mesh/unit_tests_prof-libmesh_poly2tri.$(OBJEXT) \
 	mesh/unit_tests_prof-slit_mesh_test.$(OBJEXT) \
@@ -1360,6 +1365,7 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-mesh_stitch.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-mesh_triangulation.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-mixed_dim_mesh_test.Po \
+	mesh/$(DEPDIR)/unit_tests_dbg-mixed_order_test.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-nodal_neighbors.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-slit_mesh_test.Po \
 	mesh/$(DEPDIR)/unit_tests_dbg-spatial_dimension_test.Po \
@@ -1393,6 +1399,7 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-mesh_stitch.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-mesh_triangulation.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-mixed_dim_mesh_test.Po \
+	mesh/$(DEPDIR)/unit_tests_devel-mixed_order_test.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-nodal_neighbors.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-slit_mesh_test.Po \
 	mesh/$(DEPDIR)/unit_tests_devel-spatial_dimension_test.Po \
@@ -1426,6 +1433,7 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-mesh_stitch.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-mesh_triangulation.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-mixed_dim_mesh_test.Po \
+	mesh/$(DEPDIR)/unit_tests_oprof-mixed_order_test.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-nodal_neighbors.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-slit_mesh_test.Po \
 	mesh/$(DEPDIR)/unit_tests_oprof-spatial_dimension_test.Po \
@@ -1459,6 +1467,7 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-mesh_stitch.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-mesh_triangulation.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-mixed_dim_mesh_test.Po \
+	mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-slit_mesh_test.Po \
 	mesh/$(DEPDIR)/unit_tests_opt-spatial_dimension_test.Po \
@@ -1492,6 +1501,7 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-mesh_stitch.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-mesh_triangulation.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-mixed_dim_mesh_test.Po \
+	mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-slit_mesh_test.Po \
 	mesh/$(DEPDIR)/unit_tests_prof-spatial_dimension_test.Po \
@@ -2204,9 +2214,9 @@ unit_tests_sources = driver.C libmesh_cppunit.h stream_redirector.h \
 	mesh/mesh_function.C mesh/mesh_function_dfem.C \
 	mesh/mesh_generation_test.C mesh/mesh_input.C \
 	mesh/mesh_stitch.C mesh/mesh_triangulation.C \
-	mesh/mixed_dim_mesh_test.C mesh/nodal_neighbors.C \
-	mesh/libmesh_poly2tri.C mesh/slit_mesh_test.C \
-	mesh/spatial_dimension_test.C \
+	mesh/mixed_dim_mesh_test.C mesh/mixed_order_test.C \
+	mesh/nodal_neighbors.C mesh/libmesh_poly2tri.C \
+	mesh/slit_mesh_test.C mesh/spatial_dimension_test.C \
 	mesh/mapped_subdomain_partitioner_test.C \
 	mesh/write_elemset_data.C mesh/write_sideset_data.C \
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@@ -2766,6 +2778,8 @@ mesh/unit_tests_devel-mesh_triangulation.$(OBJEXT):  \
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@@ -3798,6 +3819,7 @@ distclean-compile:
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@@ -3831,6 +3853,7 @@ distclean-compile:
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 @AMDEP_TRUE@@am__include@ @am__quote@mesh/$(DEPDIR)/unit_tests_oprof-mixed_dim_mesh_test.Po@am__quote@ # am--include-marker
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 @AMDEP_TRUE@@am__include@ @am__quote@mesh/$(DEPDIR)/unit_tests_oprof-nodal_neighbors.Po@am__quote@ # am--include-marker
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@@ -3864,6 +3887,7 @@ distclean-compile:
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@@ -3897,6 +3921,7 @@ distclean-compile:
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 @AMDEP_TRUE@@am__include@ @am__quote@mesh/$(DEPDIR)/unit_tests_prof-mixed_dim_mesh_test.Po@am__quote@ # am--include-marker
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+
 mesh/unit_tests_dbg-nodal_neighbors.o: mesh/nodal_neighbors.C
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+mesh/unit_tests_devel-mixed_order_test.o: mesh/mixed_order_test.C
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+
+mesh/unit_tests_devel-mixed_order_test.obj: mesh/mixed_order_test.C
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+
 mesh/unit_tests_devel-nodal_neighbors.o: mesh/nodal_neighbors.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_devel_CPPFLAGS) $(CPPFLAGS) $(unit_tests_devel_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_devel-nodal_neighbors.o -MD -MP -MF mesh/$(DEPDIR)/unit_tests_devel-nodal_neighbors.Tpo -c -o mesh/unit_tests_devel-nodal_neighbors.o `test -f 'mesh/nodal_neighbors.C' || echo '$(srcdir)/'`mesh/nodal_neighbors.C
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@@ -7922,6 +7975,20 @@ mesh/unit_tests_oprof-mixed_dim_mesh_test.obj: mesh/mixed_dim_mesh_test.C
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+mesh/unit_tests_oprof-mixed_order_test.o: mesh/mixed_order_test.C
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+
 mesh/unit_tests_oprof-nodal_neighbors.o: mesh/nodal_neighbors.C
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@@ -9434,6 +9501,20 @@ mesh/unit_tests_opt-mixed_dim_mesh_test.obj: mesh/mixed_dim_mesh_test.C
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 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_opt-mixed_dim_mesh_test.obj `if test -f 'mesh/mixed_dim_mesh_test.C'; then $(CYGPATH_W) 'mesh/mixed_dim_mesh_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_dim_mesh_test.C'; fi`
 
+mesh/unit_tests_opt-mixed_order_test.o: mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_opt-mixed_order_test.o -MD -MP -MF mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Tpo -c -o mesh/unit_tests_opt-mixed_order_test.o `test -f 'mesh/mixed_order_test.C' || echo '$(srcdir)/'`mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Tpo mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='mesh/mixed_order_test.C' object='mesh/unit_tests_opt-mixed_order_test.o' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_opt-mixed_order_test.o `test -f 'mesh/mixed_order_test.C' || echo '$(srcdir)/'`mesh/mixed_order_test.C
+
+mesh/unit_tests_opt-mixed_order_test.obj: mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_opt-mixed_order_test.obj -MD -MP -MF mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Tpo -c -o mesh/unit_tests_opt-mixed_order_test.obj `if test -f 'mesh/mixed_order_test.C'; then $(CYGPATH_W) 'mesh/mixed_order_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_order_test.C'; fi`
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Tpo mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='mesh/mixed_order_test.C' object='mesh/unit_tests_opt-mixed_order_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_opt-mixed_order_test.obj `if test -f 'mesh/mixed_order_test.C'; then $(CYGPATH_W) 'mesh/mixed_order_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_order_test.C'; fi`
+
 mesh/unit_tests_opt-nodal_neighbors.o: mesh/nodal_neighbors.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_opt-nodal_neighbors.o -MD -MP -MF mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Tpo -c -o mesh/unit_tests_opt-nodal_neighbors.o `test -f 'mesh/nodal_neighbors.C' || echo '$(srcdir)/'`mesh/nodal_neighbors.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Tpo mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Po
@@ -10946,6 +11027,20 @@ mesh/unit_tests_prof-mixed_dim_mesh_test.obj: mesh/mixed_dim_mesh_test.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_prof-mixed_dim_mesh_test.obj `if test -f 'mesh/mixed_dim_mesh_test.C'; then $(CYGPATH_W) 'mesh/mixed_dim_mesh_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_dim_mesh_test.C'; fi`
 
+mesh/unit_tests_prof-mixed_order_test.o: mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_prof-mixed_order_test.o -MD -MP -MF mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Tpo -c -o mesh/unit_tests_prof-mixed_order_test.o `test -f 'mesh/mixed_order_test.C' || echo '$(srcdir)/'`mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Tpo mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='mesh/mixed_order_test.C' object='mesh/unit_tests_prof-mixed_order_test.o' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_prof-mixed_order_test.o `test -f 'mesh/mixed_order_test.C' || echo '$(srcdir)/'`mesh/mixed_order_test.C
+
+mesh/unit_tests_prof-mixed_order_test.obj: mesh/mixed_order_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_prof-mixed_order_test.obj -MD -MP -MF mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Tpo -c -o mesh/unit_tests_prof-mixed_order_test.obj `if test -f 'mesh/mixed_order_test.C'; then $(CYGPATH_W) 'mesh/mixed_order_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_order_test.C'; fi`
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Tpo mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='mesh/mixed_order_test.C' object='mesh/unit_tests_prof-mixed_order_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o mesh/unit_tests_prof-mixed_order_test.obj `if test -f 'mesh/mixed_order_test.C'; then $(CYGPATH_W) 'mesh/mixed_order_test.C'; else $(CYGPATH_W) '$(srcdir)/mesh/mixed_order_test.C'; fi`
+
 mesh/unit_tests_prof-nodal_neighbors.o: mesh/nodal_neighbors.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT mesh/unit_tests_prof-nodal_neighbors.o -MD -MP -MF mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Tpo -c -o mesh/unit_tests_prof-nodal_neighbors.o `test -f 'mesh/nodal_neighbors.C' || echo '$(srcdir)/'`mesh/nodal_neighbors.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Tpo mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Po
@@ -12258,6 +12353,7 @@ distclean: distclean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-spatial_dimension_test.Po
@@ -12291,6 +12387,7 @@ distclean: distclean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-spatial_dimension_test.Po
@@ -12324,6 +12421,7 @@ distclean: distclean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-spatial_dimension_test.Po
@@ -12357,6 +12455,7 @@ distclean: distclean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-spatial_dimension_test.Po
@@ -12390,6 +12489,7 @@ distclean: distclean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-spatial_dimension_test.Po
@@ -12845,6 +12945,7 @@ maintainer-clean: maintainer-clean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_dbg-spatial_dimension_test.Po
@@ -12878,6 +12979,7 @@ maintainer-clean: maintainer-clean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_devel-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_devel-spatial_dimension_test.Po
@@ -12911,6 +13013,7 @@ maintainer-clean: maintainer-clean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_oprof-spatial_dimension_test.Po
@@ -12944,6 +13047,7 @@ maintainer-clean: maintainer-clean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_opt-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_opt-spatial_dimension_test.Po
@@ -12977,6 +13081,7 @@ maintainer-clean: maintainer-clean-am
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mesh_stitch.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mesh_triangulation.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mixed_dim_mesh_test.Po
+	-rm -f mesh/$(DEPDIR)/unit_tests_prof-mixed_order_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-nodal_neighbors.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-slit_mesh_test.Po
 	-rm -f mesh/$(DEPDIR)/unit_tests_prof-spatial_dimension_test.Po
diff --git a/tests/mesh/mixed_order_test.C b/tests/mesh/mixed_order_test.C
new file mode 100644
index 00000000000..f52c0b78430
--- /dev/null
+++ b/tests/mesh/mixed_order_test.C
@@ -0,0 +1,117 @@
+#include <libmesh/libmesh.h>
+
+#include <libmesh/elem.h>
+#include <libmesh/mesh.h>
+#include <libmesh/mesh_generation.h>
+
+#include <timpi/parallel_implementation.h>
+
+#include "test_comm.h"
+#include "libmesh_cppunit.h"
+
+
+using namespace libMesh;
+
+class MixedOrderTest : public CppUnit::TestCase
+{
+  /**
+   * The goal of this test set is to test libMesh manipulation of
+   * meshes with mixed Elem orders.
+   */
+public:
+  LIBMESH_CPPUNIT_TEST_SUITE( MixedOrderTest );
+
+  CPPUNIT_TEST( testFindNeighbors );
+  CPPUNIT_TEST( testStitch );
+
+  CPPUNIT_TEST_SUITE_END();
+
+public:
+  void setUp() {}
+
+  void tearDown() {}
+
+  static int n_neighbor_links (const MeshBase & mesh)
+  {
+    int n_neighbors = 0;
+    for (const auto & elem : mesh.local_element_ptr_range())
+      for (const auto & neigh : elem->neighbor_ptr_range())
+        if (neigh)
+          ++n_neighbors;
+    mesh.comm().sum(n_neighbors);
+    return n_neighbors;
+  }
+
+
+  void testFindNeighbors()
+  {
+    LOG_UNIT_TEST;
+
+    Mesh mesh(*TestCommWorld);
+
+    // Construct a multi-element Tri3 mesh
+    MeshTools::Generation::build_square(mesh,
+                                        /*nx=*/3, /*ny=*/3,
+                                        /*xmin=*/0., /*xmax=*/1.,
+                                        /*ymin=*/0., /*ymax=*/1.,
+                                        /*elem_type=*/TRI3);
+
+    // Make some of them into Tri6
+    auto range_start = mesh.elements_begin();
+    const auto range_end = mesh.elements_end();
+    while (range_start != range_end && (*range_start)->id() < 5)
+      ++range_start;
+
+    const int old_n_neighbors = n_neighbor_links(mesh);
+
+    mesh.all_second_order_range({range_start,range_end},
+                                /*full_ordered=*/true);
+    const int new_n_neighbors = n_neighbor_links(mesh);
+    CPPUNIT_ASSERT_EQUAL(old_n_neighbors, new_n_neighbors);
+
+    mesh.find_neighbors();
+    const int newer_n_neighbors = n_neighbor_links(mesh);
+    CPPUNIT_ASSERT_EQUAL(old_n_neighbors, newer_n_neighbors);
+  }
+
+  void testStitch()
+  {
+    LOG_UNIT_TEST;
+
+    Mesh mesh0(*TestCommWorld),
+         mesh1(*TestCommWorld);
+
+    // Quad4 mesh and Quad9 mesh
+    MeshTools::Generation::build_square(mesh0,
+                                        /*nx=*/3, /*ny=*/3,
+                                        /*xmin=*/0., /*xmax=*/1.,
+                                        /*ymin=*/0., /*ymax=*/1.,
+                                        /*elem_type=*/QUAD4);
+    CPPUNIT_ASSERT_EQUAL(mesh0.n_nodes(), dof_id_type(16));
+
+    MeshTools::Generation::build_square(mesh1,
+                                        /*nx=*/3, /*ny=*/3,
+                                        /*xmin=*/-1., /*xmax=*/0.,
+                                        /*ymin=*/0., /*ymax=*/1.,
+                                        /*elem_type=*/QUAD9);
+    CPPUNIT_ASSERT_EQUAL(mesh1.n_nodes(), dof_id_type(49));
+
+    // Stitch them together
+    mesh0.stitch_meshes(mesh1, 3, 1, TOLERANCE, /*clear_bcids*/ true,
+                        /*verbose*/ false, /*binary_search*/ false,
+                        /*enforce_all_nodes_match*/ false,
+                        /*merge_all_or_nothing*/ false);
+    CPPUNIT_ASSERT_EQUAL(mesh0.n_nodes(), dof_id_type(61));
+
+    const int stitched_n_neighbors = n_neighbor_links(mesh0);
+    CPPUNIT_ASSERT_EQUAL(stitched_n_neighbors, 54);
+
+    mesh0.all_second_order_range(mesh0.element_ptr_range(),
+                                 /*full_ordered=*/true);
+    const int new_n_neighbors = n_neighbor_links(mesh0);
+    CPPUNIT_ASSERT_EQUAL(new_n_neighbors, 54);
+  }
+};
+
+
+CPPUNIT_TEST_SUITE_REGISTRATION( MixedOrderTest );