Question regarding flipping coordinate system if z component of y-axis is negative

[NagabhushanSN95]

When transforming mipnerf-360 scene poses using PCA here, the coordinate system is flipped w.r.t. x axis if the [2,1] element is negative. As far as I understand it, when transforming a coordinate system using a permutation matrix, the extrinsics should be multiplied with permutation matrix both on the left and the right. Why is the code here multiplying only on the left?

PS: A similar operation is done to transform extrinsics from opencv convention (x,-y,-z) to NeRF/blender convention (x,y,z) for LLFF dataset poses. Since this operation is done before recentering the poses, multiplying on the right is unnecessary as it gets cancelled out while recentering. However, no recentering is done in the case of 360 dataset.

Kindly clarify.