Why are left and right reversed in a mirror but not up and down?

Our bodies are fairly symmetrical about a vertical plane, i.e. the right side is an opposite image of the left. (If we fold ourselves along a vertical line from head to toes, then each bodily side accommodates the other.) This does not apply to the horizontal plane, where the head stands in opposition to our feet.

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Another way to imagine this is to see yourself in front of and facing a mirror, but lying on your left side. You can see that your left [bottom] and right [top] sides still seem "reversed" by the mirror. You can also see that your head and feet are not reversed. A mirror image is a little like walking into a wall of soft plaster or clay, leaving a 'negative' image of yourself.

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Comment: Here is the answer (without going into too many details).

This does answer the question.

A plane mirror reverses the object ONLY in the direction perpendicular to the mirror surface. That's ALL it does.

However people often perceive this as a left right reversal with front back unchanged. That's the "psychology of perception" not physics.