Dunno if this is even worth whiting, but..
It doesn't reverse left and right in the first place. Everything that comes in, left, right, top and bottom goes back out the same place it came. However, if an object is mostly symetrical in two directions (such as humans, for instance) one tends to feel that the projection ought to be a transposed object. Transposing a human by turning it around a half turn left-right wise would sort of fit the projection, but it's local left-right would be reversed. Up/down would, of course, work similarly with a sideways human (or other object symetrical in up/down but not in left/right) - if you tilt yourself pi/2 the mirror won't magiacally reverse you chin to brow but rather reverse you in the "up/down" gravitaionally speaking from your (rather primitive) point of view.
Very good LarrrSDonald.
Here is more insight, but don't read it unless you want to know the answer.
It's a trick question. As LarrrSDonald said a mirror does not reverse left to right. It actually reverses front to back. That is to say, it reverses across the plane of the mirror. In doing so it reverses handedness and this enhances the suggestion that it reverses left to right. If you place two mirrors at right angles and look at the line of intersection you will see an image that is reversed twice and so handedness is restored. You will finally see yourself as others do.