The answer is: mirrors don't reverse left/right any more so than up/down. When you look at yourself in the mirror, your right hand side is on the right, and the left is on the left. We're more easily confused about left vs. right than top/bottom because the determination of left/right is more ambiguous than top/bottom: left/right is determined relative to the Self, whereas up/down is determined relative to the landscape. Because we are accustomed to facing others, when we face our reflection we project left/right in terms of the other's self. The point of confusion occurs as we witness and re-identify the reflection with ourselves, while hanging on to our projection to the other. Up/down is less ambiguous because we are strongly oriented by gravity, and identification of up/down is reinforced by association with distinctly different parts of the body (head/feet).
To put it succinctly, this conundrum is a confusion of identities.
This specific case is interesting because the concern is wholly a matter of language, while the presence of the mirror works as sort of a red herring that nicely leads consideration of a solution away from matters of linguistics toward physics, and as such is representative of confusions present in zillions of other highly significant matters, such as current (2004) doctrine about U.S. Foreign Policy.
[My distinction of language from physics is also an identity confusion when viewed from a certain level. ("Wowzy-wow-wow, Roxy!") suggesting important concerns for knowledge in general. See Science and Sanity]