Why do mirrors reverse images left to right? Why not up and down?

When you look in a mirror, you see a "mirror image" which (most obviously) "reverses" people's faces and writing. A common example is that ambulances have the word "AMBULANCE" printed on their front as:

so that drivers can see it correctly in their mirrors. Why do mirrors only "reverse" images in one dimension? Surely, if mirrors flip images right-to-left, shouldn't they also flip images up-and-down? How do mirrors "know" what is "left" and "right", or "up" and "down" anyway? Of course, they don't, so what is going on?

The answers to these questions are not easy, and are fraught with misconceptions. A quick search of the web turned up a few answers, all of which were ridiculously wrong. The following is my attempt to provide a correct answer. Please send me comments if you disagree.

First of all, look at yourself in a mirror. Your head is still at the top, and your feet are still at the bottom of the mirror image. Most importantly, your left hand is still on the left, and your right hand is still on the right. So, immediately, we can dispel any notion that the mirror is choosing to do some sort of X-axis or Y-axis "flip" based on "knowing" what is left-and-right or up-and-down. What has actually happened is that the 3-dimensional image in the mirror (compared to the real-world model that it is a reflection of) has been flipped along its Z-axis (i.e. the "depth" dimension). This is fundamental; this is what (by definition) a mirror does. A mirror image is a 3-dimensional model that has undergone a Z-axis inversion. Z-axis inversions do not often occur in nature (reflections in water being the only example I can think of) and therefore our brains can't easily recognize what they are looking at.

Humans have a natural left-right symmetry. So do most living things. When we are standing face-to-face with another person, it is engrained in our brains to understand, without even thinking about it, that the other person is rotated 180º (on their Y-axis) with respect to ourselves. This is how the world works. Real objects get moved in all directions (X, Y and Z transpositions) and rotated in all directions (X, Y and Z axis rotations), all of which our brains are familiar and comfortable with. When we see an image of ourselves in a mirror it is almost impossible for our brains to not assume that if our mirrored persona "turns around" 180º then they would be just like ourselves. But they wouldn't be. They are a Z-axis inverted copy of ourselves, and if they could "turn around", they'd still be Z-axis-inverted, albeit now facing away from us.

Armed with understanding these facts, i.e. that (a) mirrors display a Z-axis inverted image, and (b) human brains are determined to try to interpret mirror-images as if they are real, we can now answer the questions we started with... mathematically.

A 3-dimensional model that undergoes a Z-axis inversion, is precisely the same (mathematically) as the same 3-dimensional model that undergoes a 180º Y-axis rotation followed by an X-axis inversion. Got it? This can be proven mathematically, but it is obvious by just thinking about it. In fact, a Z-axis inversion is precisely the same as an 180º rotation along any arbitrary perpendicular axis, followed by an inversion along the remaining third perpendicular axis.

Whenever we look in a mirror, our brains generally make the most sense of what we see (a Z-axis inverted image) by imagining that the object is rotated by 180º, along the Y-axis, with respect to our eyes. And that's correct, but we must also then see the corresponding X-axis flip. And we do, in such details as people's faces, and in writing being "reversed".