(Note that this question is migrated from the physics SE... I apologize for the imprecise language)
The only $Z_2$ symmetries I can think of the torus are reflection on plane, whose quotient should form an open cylinder, and rotation by 180 degrees, which becomes the "branched covering" of the 2-sphere with four "branch points" or fixed points. I do not understand exactly what the branch points are, they seem connected to the think that physicists call "cone singularities", but I'd guess they are not the same object.
My questions are:
-is there some other orbifolding of $T^2$ under $Z_2$?
-are not both orbifolds simply what should happen if you try to impose a flat geometry to a sphere? The former one, the cylinder, looks very much as the sphere removing N and S poles and then using a flat metric of paralells and meridians. The later looks more sophisticated, with four special points, but still could be the same trick, is it?