When you look into a full-length mirror and raise your right hand, your image raises its left hand. And yet both your heads are still on top. So how does a mirror reverse things right to left, but not top to bottom? Does the mirror somehow know horizontal from vertical?
This is one of those loaded questions that can drive you crazy because the question itself is deceptive. It starts with a mistaken assertion and asks you do your reasoning from that point. But you can't pursue the road to truth if you are tricked into starting off in the wrong direction.
A mirror does not reverse things from right to left. It reverses things from front to back or, you might say, from away to toward.
Stand in front of a full-length mirror. Let's name the person in the mirror Egami. Now how do you think Egami got that way, with his left arm toward your right and his right arm toward your left? I'll bet you seven years of bad luck that you think Egami got that way by rotating your body half a turn, executing an about-face. Well, that's exactly what you did -- in your imagination. When you look at your reflection, it's you that imagines yourself rotated. The mirror didn't do it. All a mirror can do is reverse a direction. It can't rotate anything, because it's strictly two-dimensional.
Your mirror took its incoming light and reflected it straight back at you. From your perspective, the light went away from you and was sent back toward you. The mirror reversed the direction of the light, not the orientation of any image. So Egami is simply you with your toward and away directions reversed. You are, of course, in the habit of looking away from yourself, but Egami is looking toward you, and whenever a person is facing you and looking toward you, his left arm will be on your right, no? What's so unusual about that? No rotations or right-to-left swaps required.
Note that the words "up" and "down" appear nowhere in the foregoing. They're completely irrelevant to two people -- or one person and his image -- who are facing each other. Up and down mean exactly the same thing to both of them. No difference. (Unless, of course, one of them is standing on his head.)
But if we wanted to, we really could get a mirror to reverse up and down. Just hold a mirror high above your own head, parallel to the floor, and look up. You'll see that Egami is now standing on his head. As expected, the mirror has reversed its toward and away directions, which from your current viewpoint just happen to be up and down. You can get the same effect by placing a mirror on the floor. Again, you'll see Egami upside down. (Caution: don't step on the mirror.)
Now look into the bowl (the hollow part) of a polished, shiny spoon. You'll see that your image is upside-down, no matter how you move the spoon. How does it do that?
The spoon's inner surface is concave -- that is, it is hollowed away from you, like a cave. (That's a good way to remember the distinction between concave and convex.) Note that the top part of the spoon's surface is tilted downward, sending its reflected light downward, just as the overhead mirror did. And the bottom part is tilted upward, sending its reflected light upward just as the floor mirror did. Thus, the reflected image has its bottom up and its top down. Your face has been turned upside down.
Now look into the outer (convex) surface of the spoon and you'll see that your image is right-side-up. That's because the shiny surfaces are all tilted outward, reflecting their light in all directions away from your face. So your image isn't displaced in any direction and looks normal, except for distortions due to the curvature of the spoon. Fun-house mirrors make you (or your Egami) look all sorts of grotesque ways because their mirrors are purposely curved in all sorts of grotesque ways.
Oh, and about that name, Egami? That's your Image, spelled from right to left.
Adapted from What Einstein Told His Barber. See much more here.