Imagine writing a message on a transparent sheet of celluloid. Now stand in front of a mirror, and hold the celluloid up so you can read it normally.
Now turn the celluloid around and show it to the mirror. Look at the mirror image, and note that it is backward, just as you would expect, just like when you hold a book up to the looking-glass like Alice did.
But wait! Since the celluloid is transparent, you can see the message there by looking directly at it! And what do you see? Looking directly at the celluloid, it is backward! Now look at the mirror. It’s backward too. So what you see directly in the celluloid is exactly what you see in the mirror. It’s not backward at all!
Well, I mean, it is backward in the mirror, but it’s backward when you look directly at (the back side of) the celluloid too. So they’re exactly the same! The fact of looking at it in the mirror isn’t reversing anything, nohow.
So why does the text read backward – both in the mirror and looking directly at the celluloid? It got that way when you turned the celluloid around to show it to the mirror! – That’s what reversed it left-to-right!
Unless, of course, you turned the celluloid around to show it to the mirror by flipping it top-for-bottom. Then it’s reversed top-to-bottom. (Try it!) But not because the mirror is doing it. It’s because you actually did flip the celluloid over that way.
(ETA: So much for the OP’s original question, I guess.)