Question

By asking why the mirror reverses left/right but not up/down, you are not comparing apples with apples. "Up" is a straightforward direction to define. It is simply representable by an arrow pointing in some chosen direction, such as away from the earth's center. Or if we are floating far from Earth, we might take "up" to be the direction from our feet to our head. The same goes for in/out: we can take the "in" or "front" direction to be the direction pointing away from the front of our body. If you wanted to tell a Martian which way is "up" and which way is "front", you could do it easily.

"Left" and "right" are not so easily defined, and you'd have a much harder time telling the Martian what you mean by these. That's because "left" and "right" are relative: they depend on our deciding on an "up" and a "front" first. So comparing "right" with "up" is not a fair comparison.

I think "left" and "right" have more to do with the bilateral symmetry of our body. We have all kinds of words for one-dimensionally defined directions: up/down, front/back, north/south/east/west. They are just fine for explaining where things are. They all, of course, need a certain reference (e.g. the earth, or our body) for their definition, but they are all still one-dimensional. But trying to explain which hand to reach out for shaking hands with somebody--the "right" hand in most parts of the world--will only work when the directions "up" and "front" are already defined. We have invented the words "left" and "right" because we needed them quite often to distinguish between the two orientations--of course not only referring to the two sides of our body, but also all kinds of other things with two sides. It's not a coincidence that mathematicians also use these words for distinguishing between three-dimensional co-ordinate systems being "left handed" or "right handed" in their orientation. There are only these two different possibilities for joining three arrows together to form a perpendicular set of axes. No matter how you rotate them in space, they can never "cover" each other, since they are mirror images of one another, just like our hands are.