The answer
Q: Why does a mirror reverse your reflection side-to-side, but not head-to-toe?
A: It doesn't.
The physics formula for reflection is:
angle of incidence = angle of reflection.
A reflective surface (i.e. a mirror) doesn't know anything about what it's reflecting. The light waves bounce off according to the above formula regardless of where they originate.
Why then does your reflection appear reversed? The answer to that lies in human psychology -- the high value we place on symmetry. Our bodies are not perfectly symmetric in any way, but they are much more symmetric left-to-right than any way. Therefore, when we mentally compare ourselves to our reflections, we automatically choose to rotate around the head-to-toe axis. This mapping preserves the most symmetry. When compared to the reflection that way, we perceive it to be reversed left-to-right.
That's not the only mapping though, and there's no reason (other than symmetry) to prefer it. If you imagine flipping your body upside down, as if you pivoted around a horizontal bar, then you'd say that the your standing reflection was reversed head-to-toe rather than left-to-right.
If you pictured yourself rotated around a line sticking out of the mirror instead, then you'd say that the mirror reversed your reflection both ways.
Finally (and this is the hardest one), if you can imagine pushing your backside out through your front without rotating at all, and comparing that transformation to the reflection, you'd see that the mirror didn't flip your image at all.
So, this teaser wasn't about physics at all, but about human perception.
I hope that was interesting,
Eric
