Alternatively, the basic reason is that
mirrors reverse front and back. Our
notions of
left
and
right describe the
relation between "this side"/"that side" and front/back. On the contrary,
top and
bottom are
independent of the front/back
direction. In the mirror image, "this side" and "that side" are unchanged, just like top and bottom, but front and back are swapped, so in effect left and right are reversed.
Interestingly, a combination of two mirrors at a right angle does not reverse left/right. In addition to front/back, it reverses the sides. The net effect is like (-1)*(-1) = 1, and the notions of left/right are unchanged.