What a mirror does is reflect light with respect to the surface normal at the point the photon hit it. The surface normal is basically the direction sticking straight out of the mirror (technically, it’s the normal of the cross product of the two vectors that describe the plane of the mirror). Since we have depth perception we will see the reflected light as coming from beyond the mirror, even though it is coming from behind us. The mirror really does just invert the viewing frustum. Nothing is different. Nothing is on a different side.
So if it doesn’t actually reverse left and right, why does our brain think it does? We look at ourselves in the mirror and our brain assumes it’s another person looking at us. That’s when one’s mind’s sense of depth perception plays tricks on one’s sense of direction. Since our brain is saying that person is looking at us, we think that left and right directions are different when they’re not. In other words, we just confuse ourselves because we’re not used to mirrors! Our visual system never evolved to take a mirror into account. So we have to learn it. That’s why when we’re driving and someone puts on a turn signal and we see it in the mirror, we know that they are still turning in the same direction as the side of the car their turn signal was blinking. It’s the same psychology that makes driving a car backwards awkward at first.
Then why do images and letterings appear backwards? I’m glad you asked. When one looks at the writing in a mirror one expects its orientation to change just like one thinks a person (who is mostly symmetric, while text is not) is facing the other direction. Automatically, one reads it backwards. Normally, the text’s right is on one’s left. If one is holding the text so they are facing the same way, it will appear backwards to them.
It’s the same reason everything on a storefront window display looks backwards once one is inside the store. If you were out side the store and looking in and there was a mirror inside reflecting the writing in the window, it would be perfectly readable to you in the mirror as well as in the window. If you were between the window and the mirror, you could read the writing in the mirror, but the writing in the window would be backwards to you. Since mirror takes what behind you and puts it in front of you, you can read what is behind you through the mirror as though it was in front of you, just as though you were outside looking in. Another off the wall thought is that “Ambulance” is written out backwards — not just so that one can read it in you rear view mirror — but so that if the ambulance were transparent you could read the “backwards” writing in its front from behind the vehicle with the text oriented the right way.
Basically, a mirror universe operates exactly the same way ours does, but exactly opposite of how our brain wants to think it works.
Aren’t mirror’s wonderful!?