You see the world because light gets into your eyes. You see these words, for example, because light from the screen or reflected off the printed page enters your eyes and makes an image on your retina.
When you make a visual picture of the world, you assume that the light entering your eyes has traveled in a straight line to reach you. But mirrors and other shiny objects change the path of the light, bouncing it back in an organized fashion. When you look into a mirror, you see your image because light reflecting from your face bounces off the mirror and back into your eyes (click to enlarge the diagram below). Your eyes and brain assume that the light has traveled in a straight line to reach your eyes, so you see an image of your face out there in front of or behind the mirror.
When you look into an ordinary flat mirror, the image of your face is right side up: Your hair is on top of your head and your chin is underneath. To reach your eyes, the light from your hair hits the mirror at a slight angle and then bounces into your eyes from above—which is why you see your hair on top and your image as right side up.
When you look into a cylindrical mirror with the axis of the mirror horizontal and with your face a foot or more away from the mirror, your image is upside down. That’s because the light from your hair bounces off the curved mirror and comes to your eyes from below.
To make sense of the angle at which the light is entering your eyes, your eyes and brain must see the image of your face as upside down and a little bit in front of the mirror (click to expand diagram below).
As everyone knows, a flat mirror reverses your right side and your left side. How does it do that? Suppose you are standing face to face with someone. If your right ear points toward the east, the other person’s left ear will point toward the east. Now, instead of facing another person, suppose you are facing a flat mirror with your right ear pointing to the east. The light from your right ear will bounce off the flat mirror and enter your eyes from the east. Even though your east ear is the east ear of the image, your right ear has become the left ear of the image! (Yes, this is a little mind-boggling at first reading. But once you get it, it will seem simple.)
Now look into the cylindrical mirror with its axis vertical. Stand at least a foot away from the mirror. Once again, place your right ear so that it points to the east. Light from your right ear bounces off the curved mirror and enters your eyes from the west. Light from your right ear appears to come from the right ear of the image. In this cylindrical mirror, you see yourself as others see you. You see the image of your face just a little bit in front of the mirror.