Why do mirrors reverse left and right but not up and down?
They do not. A mirror reverses neither — it reverses front and back, along the axis pointing into the glass. Everything else is an artefact of how you describe what you are seeing.
Consider what a mirror actually does to a point in space. Your nose, closest to the glass, appears closest in the reflection. The back of your head, furthest away, appears furthest. Left stays on the left side of the room and right stays on the right; up stays up. The only coordinate that flips is depth. A mirror image is the original with one axis inverted, which is why it cannot be rotated to match the original — the same reason a left glove cannot be turned into a right glove.
The confusion comes from the words. "Left" and "right" are defined relative to the body they describe, whereas "up" is defined by gravity and stays fixed no matter which way you turn. When you look at your reflection, you unconsciously imagine yourself turning around to face the other way. That imagined rotation is what swaps left and right, and you attribute it to the mirror.
Two demonstrations settle it. Lie on your side facing a mirror: the reflection now appears to swap head and feet, not hands — because you have changed which axis your body description is aligned with. Or hold up printed text: the letters are reversed left to right only because you rotated the page to face the glass. Hold the page facing you and read it through the back, and it is not reversed at all.
The mirror behaves consistently. It is our habit of describing reflections as rotated copies of ourselves that introduces the apparent asymmetry.