It assumes that the mirror's face is its front, making the right side of the mirror near your left hand. I only described it this way to reduce word count. We can say something similar using your more precise terms-- "near left/right hand".

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.