Why do mirrors flip images side-to-side but not top-to-bottom?

This has been the subject of an informal discussion between a couple of people in the office today who are (obviously) not wholly competent to answer it.

When we look in a mirror, from experience we know that “mirror right” represents “my left” and vice versa. That is, the image appears on its face to have been “flipped” from left-to-right and right-to-left. Granted. We all get that; we see it every day.

So the question is: Why doesn’t the image ALSO appear to be flipped top-to-bottom and bottom-to-top?

Our discussion got pretty lively, and was not completely resolved to the satisfaction of the other party (though he seems to think so). I’m satisfied with what I understand now, so I’m in no particular need for an answer – and not even interested enough until now to google the answer. I know what I know, and I’m in no more mystery. But it seemed like an interesting question, so there you go.