An old chestnut asks why mirrors reverse the directions LEFT and RIGHT, but maintain UP and DOWN.

Actually, it's not a question about mirrors, but a question about the different concepts we have of direction. Think of yourself standing in front of a mirror. Your left hand is opposite your image's right hand, your right hand is opposite your image's left hand, your head is opposite your image's head, and your feet opposite your image's feet. You could equally well look at somebody else, and ask why your head and feet are opposite theirs, but your left and right hands are reversed compared to theirs. So let's do that, instead.

Now look at how you transfer your notions of directions to the other person. To transfer left and right, you pretend to rotate yourself by 180o about a vertical line between the two of you. Naturally, after rotation your left goes to the point opposite your right.

Suppose you wanted to transfer you up and down in this manner. You'd pretend to rotate yourself by 180o about a horizontal line between the two of you. Lo and behold, you're now standing on your head, with your feet (after rotation) in front of where your head started off. But that's not how you transfer these directions; instead, you use the same rotation, about the vertical line, for all 3 directions

Your front is also opposite the other person's front. It's all due to the screwy notion we have of how to transfer left and right!