Why does a mirror reverse left and right, but not up and down?

Stand in front of a mirror and note that the positions of your left hand, right hand, head, and feet all correspond with their reflections. A mirror does not reverse up, down, left, or right! It reverses the direction of near and far, relative to yourself.

The misconception that mirrors reverse left and right arises because when we stand and turn to face other people, our left and right sides are reversed from theirs. Gravity compels us to orient about a vertical axis, and bilateral symmetry makes our left and right sides look similar. As a result, the common sense belief is that the everyday reversed appearance of other people is the only valid one, and when we look at ourselves in a mirror, all we notice is that our left and right sides have been swapped.

If we always looked at other people by standing on our heads, their upside-down appearance would seem normal. Thus, even with the lack of head to feet symmetry, our own reflection in a mirror might appear to reverse top and bottom, but not left and right.
[Photo courtesy spaceflight.nasa.gov]

This special corner mirror reverses both front to rear, and left to right.