I guess I should state the full question:
Why do mirrors reverse left and right, but they don't reverse up and down?
Answer: Mirrors don't reverse left and right and they don't reverse up and down. Wouldn't that be kind of funny if I just stopped here? But you know I can't.
So, here is an image of Obi-Wan standing in front of a mirror. You know, even Jedi have to check their hair. Yes, I "borrowed" this from one of my kids.
What should you see here? I have added some arrows to point out that the image of the top of Obi-Wan's head is at the top of the mirror. The image of Obi-Wan's right hand is on the right side of the mirror. There is no switching. Why is there a problem?
I think the problem is that if you were on the other side of the mirror (and not just an image), your right hand would be on the left side of the mirror. However, your head would still be at the top of the mirror. Well, this is true if you walked around to the other side. Here is an image of Obi-Wan (a clone of Obi-Wan - let me call him Obi-One) walks to the other side of a partial mirror.
In this case, the top-bottom of Obi-One is the same top-bottom as the image of Obi-Wan.
Ok. But Obi-One (although a clone) is still a Jedi. What if he wanted to go to the other side of the mirror by going over the top? Here is a picture of that.
In this case, Obi-One and the image of Obi-Wan have right hands on the same side. However, their top and bottom are reversed. So, now we can see the problem. The problem of the mirror comes from our cultural background. We assume that the image is like someone walking around to the back side of a mirror. If we assumed that people would get to the back side by going head first over the top, I would be answering a different question. That question would be: Why do mirrors reverse up and down?
There is something that mirrors do flip. They flip front and back. Just imagine that you were Obi-Wan and you walked forward a bit (to where the image would be). Your right hand would be on the right side still. However, your back would be facing towards where the mirror was, not your front. So, mirrors flip front and back.
