Let the mirror be at the end of a hallway leading north.
You face the mirror.
Your mirror image faces south. This makes sense, because the mirror
faces south, and so the direction _perpendicular_ to the mirror is
special.
Your right hand, which is on your east side, is also on the east side
of your mirror image...
just as your head, which is on top of you, is on the top of your
mirror image.
So, considering you and your mirror image in absolute coordinates,
neither up and down nor east and west are reversed: north and south
are reversed, which are special, because that axis is the one
perpendicular to the mirror, the other two being parallel to its
surface.
Normally, though, you don't talk about your "east" hand or your "west"
hand. Instead, you talk about your right hand and your left hand.
Right and left aren't absolute directions, the way up and down (or
east and west) are; instead, they are relative directions. They're
defined in terms of up and down and front and back (look at an object
from its front - hence, you are looking in its relative "back"
direction: right is 90 degrees counterclockwise from up, left is 90
degrees clockwise from up).
So, when you are facing the mirror, what happens makes sense: up and
down stay the same, but since north and south, corresponding to front
and back, are reversed in your image, the relative direction "right",
for your image, points to the image of your _left_ hand.
Suppose you weren't facing the mirror, but looking sideways. Then
right and left would be reversed in a "real" sense, but that wouldn't
make a difference: one still sees right and left as reversed, because
one maps oneself on to one's image by performing a rotation with a
constant vertical axis. Because that's a kind of motion people do in
real life more often than, say, backflips.