This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
Why does a mirror appear to invert the left-right directions, but not up-down?
Mirrors invert front to back, not left to right.
The popular misconception of the inversion is caused by the fact that
a person when looking at another person expects him/her to face her/him,
so with the left-hand side to the right. When facing oneself (in the
mirror) one sees an 'uninverted' person.
See Martin Gardner, ``Hexaflexagons and other mathematical
diversions,'' University of Chicago Press 1988, Chapter 16. A letter
by R.D. Tschigi and J.L. Taylor published in this book states that the
fundamental reason is: ``Human beings are superficially and grossly
bilaterally symmetrical, but subjectively and behaviorally they are
relatively asymmetrical. The very fact that we can distinguish our
right from our left side implies an asymettry of the perceiving
system, as noted by Ernst Mach in 1900. We are thus, to a certain
extent, an asymmetrical mind dwelling in a bilaterally symmetrical
body, at least with respect to a casual visual inspection of our
Martin Gardner has also written the book ``The Ambidextrous Universe.''