After all, a mirror doesn’t know or care which way is “up” — in fact, there is no such thing as “up” built into the world. It’s just a term we have for a direction pointing away from some plane we’ve designated as the “ground,” and we only bother to define these concepts because we humans happen to be very concerned with gravity and its effects on us. The light bouncing off of a mirror, one assumes, has no such concerns — it could care less about our concepts of verticality and horizontality. But then why is there this asymmetry in the way a mirror reflects our image, turning us around left-to-right but not top-to-bottom?
The first step towards an answer is to realize that the question is flawed. The mirror doesn’t actually reverse your image either left-to-right or top-to-bottom — it reverses your image front-to-back, that is, along the axis perpendicular to the mirror. Imagine you had a hollow Halloween mask, and you turned it inside out. That’s exactly what a mirror does: it “turns you inside out,” so that you’re facing the opposite direction without having been rotated.
But if the mirror is just flipping our image front to back, why does it look like we’re being flipped left to right? It’s because the left and right sides of our bodies are almost identical. So the inside-out person you see in the mirror looks a lot like what you would see if someone created a clone of your body and rotated it 180 degrees so it was facing you. That’s why you have the strong feeling that the mirror is rotating you in the horizontal plane, even though it’s actually just turning you inside out.
The illusion wouldn’t be nearly as strong if our bodies didn’t have left-right symmetry. Let’s say you had a tentacle in place of your right arm. Then your mirror image wouldn’t look like a 180-degree rotation of yourself, because the tentacle would be on the wrong side. The mirror’s ability to make us feel like an image has been turned around only works with a symmetrical axis.
Another experiment you can do to drive this point home is to try lying down on the floor, on your left side, facing a mirror. Now your mirror-image will be lying on his right side with his left side on top, while you’re lying on your left side with your right side on top, creating the illusion that the mirror is flipping your image vertically, as opposed to when you were standing in front of the mirror and it looked like your image was being flipped horizontally. This proves that the nature of the illusion is different than we’d originally thought: it’s not that the mirror seems to rotate you horizontally rather than vertically, but that the mirror seems to rotate you around your symmetrical axis rather than your asymmetrical axis.