If I stand in front of a (horizontally) concave mirror at a distance equal to twice the focal distance, I should be able to see my reflection inverted. That is, if I lift my left hand, the reflection will lift the hand to my right. So basically it would be like a normal flat mirror, except that it doesn't invert left and right.
Now imagine that I can change the curvature of the mirror, making the focal distance progressively bigger, but I stay at twice the focal distance, when the focal distance becomes infinite (also placing me at infinity), it will have become a flat mirror but that doesn't flip my image.
So basically this thought experiment tells me that if you stand infinitely far in front of a flat mirror, your image won't be reversed. Is this physically plausible, a mathematical artifact/loophole or just plain stupid?