Why does a mirror show you back to front but not upside down?

Physics-like answer: There is no flipping at all. Each point on your body is reflected back from the mirror at the location where where the light from that part strikes the mirror.

E.g. draw a "ray" from your left hand to the mirror. It strikes the left side of the mirror and comes back to your left hand. (And another ray comes to your eye, but ignore all but the perpendicular rays.)

Linguistics-like answer: It's because we name our hands etc with directions left and right, but not our heads vs feet.

If we named our hands "writer" and "puncher" we might not suffer from this riddle.

EDIT: Stand in a relaxed hands at your side position. Label the top of the mirror "top," and the bottom "bottom," relative to gravity.

Your head, at the "top" of your body relative to gravity, is at the top of the mirror.

Now pierce your body through the belly button with an axle, and rotate your body 90 degrees. Your left hand is at the "top" of your body, relative to gravity, and so is its reflection in the mirror. No flipping.

FURTHER EDIT: Label the mirror at the top with an arrow pointing up, and the word "north." Three more arrows, with east west and south.

Call your left hand "west." Wave your west hand. What do you see? No flipping.

YET ANOTHER EDIT: Neurologic-like answer. ... I don't know how to do this one, but I suspect there is one.

A classic puzzle, the answer to which is that the projection takes place along the z-axis rather than x-axis.

Richard Feynman nicely explains this in this video: https://youtu.be/6tuxLY94LXw

They don't reverse left and right. They reverse forward and back.

Things on the left side of your body are still on your left as you look at the mirror. You imagine switching places with the mirror image by turning on an axis down your body, because that's how you're symmetric. That also reverses forward and back, but not the way a mirror does, so you get an extra left-right reversal as well.

If you had writing on your chest, it appears reversed in the mirror because it's reversed in front of you. If you looked through a tee-shirt, you'd see that the things seem reversed. The writing is meant to be read by a person looking at you, not by you. The first letter is on your right, so that it's on the left to a viewer in front of you looking towards your back.

Because left / right are relative and up / down are absolute.

If you want horizontal absolute coordinates, try north / south / east / west.

If you want relative vertical coordinates, try lying on your side.

Why Do Mirrors Flip Left & Right (but not up & down)? - https://www.youtube.com/watch?v=1t4dOPxKgrY

Mirrors only flip in and out, not left and right or up and down.