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

Conclusion: A mirror does not actually reverse left and right. It only reverses front and back.

Only the direction perpendicular to the mirror surface (depth) is inverted.
For example, in coordinates, when the mirror is in front of you:
[
(x, y, z) \rightarrow (x, y, -z)
]
Left–right ((x)) and up–down ((y)) stay the same.

Why it looks like a “left-right reversal”:

1. We interpret our mirror image as “another person facing us.”
2. To compare with that person, we mentally rotate ourselves by 180° (usually around the vertical axis).
3. That rotation swaps left and right.

So the mirror itself is not swapping left and right; our way of comparing is what creates that left-right swap.
Likewise, if you rotate around a different axis before comparing, you can make it appear as if up and down were reversed instead.