Why does a mirror exchange left and right but not up and down?

A lot of people seem to ask this question when they wear a T-shirt in front of a mirror. For example, imagine wearing a T-shirt with lettering on it while brushing your teeth. Why are the letters on the T-shirt reversed in the mirror, while your head appears right side up? Why does a mirror only reverse things in the horizontal dimension? Go stand in front of a restroom mirror and ponder this for a moment...

For example, here is a photo of a person wearing a How Stuff Works T-shirt while standing in front of a mirror. The front of the T-shirt reads "I Gotta Know...", and you can see that the mirror appears to reverse it:

One way to start to get a handle on this is to take a piece of paper with your name written on it. Look at it in the mirror and it will be reversed. Now turn it vertically. You will find that the lettering is reversed, but it is not upside down. That should start to give you a clue as to what is going on. Now take your finger and point to the first letter in your name, both on the paper and on the mirror. Notice that there is a one-to-one correspondence -- if the letter appears on the left hand side of the paper from your perspective, it appears on the left hand side of the mirror.

Take a piece of thin, translucent paper and write your name on it. Stand in front of a mirror and hold the paper up so that you can read the paper normally. Now look in the mirror. You are seeing the back of the translucent sheet in the mirror, and the word is not reversed -- it looks completely normal. Now turn the paper over and look at it in the mirror. It is reversed, but so are the letters on the back of the translucent sheet. Note that you turned the paper over -- you reversed it!

In other words, the reason why letters look reversed in a mirror is because you are presenting them to the mirror reversed. The reason why someone coming toward you can read the message on your T-shirt is because the letters on your T-shirt are reversed (from your perspective) when you wear it. If you printed the words on the T-shirt so that you could read them (from the inside of the shirt), the words would look reversed to an approaching person but they would look great in a mirror.