Mirror, Mirror, on the wall...

I was looking in the mirror today, as I am wont to do, and I wondered:
Why do mirrors reflect right-to-left, but not up and down?
Pinhole cameras flip things vertically; so do some funhouse mirrors. One side of a spoon reflects upside-down, the other right-side-up. How do mirrors work?
I asked a neurologist.
The answer surprised me. He said that mirrors don't reflect right-to-left at all, they act on the distance dimension.
After thinking about this for a while, I came up with an example which helped me understand it.
Imagine you are putting on a mask. You see the inside (concave) surface, shown here on the left:








What you see in a mirror is yourself, but ahead of you. It is as if you pulled the mask inside-out: the right side of your face (in green) is still on the same side, but further ahead.
If you looked at someone else wearing the mask instead of your reflection, you would see the facade on the right.
The illusion of rotation is seductive because our faces are symmetrical. If our heads lay flat instead of vertical, we'd see much more clearly.