Why do symmetrically identical edges bevel differently? I have a symmetrical object, the base of a column, and when I attempt to bevel the four corners of this object, each corner bevels by a different amount. 



When I rotate the mesh about the z axis, the same edges continue to be beveled by the same inconsistent amount, so this difference has to be coming from the mesh somehow. 

I can to some degree ameliorate the problem by deleting all but one of these edges and recreating the mesh by extruding and rotating by 90 degrees about the origin a few times. This greatly reduces the unevenness of the bevel, but not entirely.

Is the mesh just imperceptibly inaccurate, and if that's the case, why does such a small inaccuracy lead to such a large disparity in the bevel size?
If you go to the options for he bevel operation (the toolbar or press f6) then you can change how the width is determined. In your case since the edges are equal length all around percent would work. I've noticed the bevel tool behaving weirdly as well (specifically when offset is used). Maybe it's a bug in the latest version of Blender.