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Mirrors can totally reverse top-to-bottom, you just have to bend over to see it. The left-right bias is based on the way we look behind us, not any property of the mirror.
This takes a little explaining.
A rotation is a reversal through two dimensions at once.
If you turn around to look behind you, you're swapping front-and-back, AND left-and-right.
If you stand on your head, you're swapping front-and-back AND top-and-bottom.
Stand facing the way the mirror does, then turn to look into it. You have to do some kind of rotation - a two-dimension reversal - to get there. If you're a normal human, you'll twist around, swapping left-and-right as you swap back-and-front. Your left and right ear swap places, your nose and the back of your head swap places too.
But your reflection doesn't do that.
A mirror only reverses ONE dimension: front-and-back. It's the equivalent of punching your face out the back of your head: its ears are still on their original sides. You have swapped left and right in order to face in the opposite direction, but your reflection hasn't - so it's ears are on opposite sides to yours.
But you can do it the other way.
Stand with your back to the mirror, and bend over and look under your arm (or between your legs) to see your reflection, instead of twisting around.
Hold something with writing on it, and you'll see: the letters in the reflection are upside-down, but they face in the right direction.
The only reason you don't see this very often is that it's a fucking weird thing to do and nobody ever does it.
Just look at the reflection of the sky in a lake (try any Bob Ross painting) to see a mirror reflect top and bottom