the call to ax.invert_xaxis() (and the y-axis equivalent) blindly swaps the limits using set_xlim(). This sets autoscale to False. I think the autoscaling attribute should be left unchanged so that one can use invert_axis() prior to any plotting and still have the direction of the axis maintained. If others agree, I can quickly make a PR that fixes this in Axes and Axes3D.
Added a test for #1553
Closed by #1557.