"A man is shaving, with his razor in his right hand. He looks into his mirror and sees his image shaving with his left hand. He says, "Oh. There's been a reversal from right to left. Why is there no reversal from top to bottom?"
Left and right aren't reversed (or perhaps I should say left and right are impossible to define properly). My hand holding the razor is to the West. It's the one furthest from my heart, which is on my left. The same goes for my mirror image. The reality is that a mirror reverses back to front. While I face South, my image appears to face North. It puts whatever part of me is closest to the mirror, also closest in my image. Toward and away from the mirror becomes the coordinate system I share with my image, rather than North and South. My logical mind tells me that in the real world I cannot reverse back to front without also reversing left to right or top to bottom, so I try to define my image's right hand as it's left.
Consider the problem of communicating with a being on a planet circling a distance star (hopefully with faster-than-light transmissions). How would you explain which side of your body your heart is on, which hand you use the most, or which way the Earth rotates? You would have to rely on mutually recognized "landmarks" in asymmetrical surroundings in order to define direction. Assuming the other being was in our galaxy, probably the Andromeda galaxy would be good for a first landmark, but more would be needed. But if "he" could be anywhere in the universe, defining left and right would be virtually impossible.