Assuming a mirror's face is its front: Photons bounce off the right side of you, strike the left side of the mirror, and then reach your eyes. Similarly for the left side. Photons bounce off the top side of you, strike the top side of the mirror, and then reach your eyes. Similarly for the bottom side.
I think this explanation is based on the (erroneous) tendency of people to think of what they see in the mirror as themselves turned around. Look at the desk in front of you. The right hand drawer is near your right hand. The left hand drawer is near you left hand. Now why did you just call the right side of the mirror, which is near your right hand, as the left side?
I did not describe the mirror "turning you around". If it was yourself turned around, you'd see photons from your doppelganger's right hand near your left (as is the case when you face someone). Since it's a reflection, which does not turn anything around (as you said), your image of your left hand is near your actual left hand. To wrap it all up-- the reflection of your head is near your head, and your feet is near your feet. This is why you are reversed left-right, and not top-down in a flat mirror.