Several years ago I was listening to the Feynmann Lectures and someone asked why mirrors reverse left and right but don't reverse up and down. I never heard Dr. Feynmann's response but I imagine it would have gone something like this: The problem here is poor choice of words, we actually have 2 meanings of the word "left". One definition is handedness, if a corkscrew digs into the cork when twisted counterclockwise it is a left handed corkscrew. A left handed glove will not fit a right hand.
The second definition is best described using the anatomical word "mediolateral". This is a direction, and is not in any way related to handedness. Mirrors reverse handedness, they only reverse handedness, they do not reverse directions. The left direction is not reversed any more than the up direction is reversed.
Because up, down, left, and right can all be rotated into each other they are all equivalent and a mirror does not break that equivalence.