Mirror's don't flip left and right. We implicitly compare a mirror image to the original by rotating the object, and naturally we maintain top/bottom orientation, this gives the appearance of "flipping" the object left to right. However, if you use vertical rotation to super-impose the object on its mirror image you'll see that it seems to have been flipped top-to-bottom instead.
The way I see it, the mirror does flip up and down, at least by the measure of the same experiment used to prove that it flips left and right.
To prove that it flips left and right, imagine a vertical pole between you and the mirror. The pole has a card attached to it with the front facing the mirror. The two dots, red and blue are painted on the card and you can see their reflection in the mirror.
In the reflection, the blue dot is on the right and the red dot is on the left.
If you rotate the pole so that is facing you and you can now see the front of the card you see that the blue dot is really on the left and the red dot on the right.
This proves that the mirror flips left and right.
To prove that it flips up and down, imagine the same pole in front of you with the card facing the mirror, but this time the pole is horizontal.
in the reflection you see that the blue dot is at the top and the red dot is at the bottom.
If you rotate the horizontal pole so that the card is now facing you, you see that the blue dot is really at the bottom and the red dot at the top.
"If you rotate the pole so that is facing you and you can now see the front of the card you see that the blue dot is really on the left and the red dot on the right.
This proves that the mirror flips left and right."
All it proves is that you've rotated the card. You've done the flipping, not the mirror.
Now, without doing any rotation to the card, if the card is seen from the perspective of the mirror, the blue dot is on the left, and the red on the right.
Depending on which perspective you're looking at an object from, one side will be on your left, and the other on your right. But no flipping takes place, unless you either move yourself to a different perspective, or rotate the object. A mirror does neither. It just reflects back exactly what's in front of it.. so you see whatever's on your right on your right, and whatever's on your left on your left. There is no flipping.
The effect of left/right "flipping" is because we are left/right symmetrical and so we can imagine ourselves being the person in the mirror by turning that way. For up-down flipping to happen we have to be top-bottom symmetrical to even begin to think about it, although it might not be that simple since gravity continues to differntiate those directions.
Another way of thinking about it is mirrors swap what is near and far instead of rotating. If your twin was standing on the other side of glass without rotation you would see their back instead of your front. If the mirror is above you then you see your head. If you twin is standing above you then you see their feet.
Flipping thins inside out sounds strange, but it also works. If you take a mask with the same face on the inside and outside and look behind the mask you see the same thing as what you see in the mirror excluding depth perception issues.
Finally, you can think of mirrors as swapping both left and right and top to bottom. If your twin is rotated so their right hand is touching your right hand and their left hand is touching your left hand then their feet are up in the air. (Easiest to visualize with two action figures.)
This works for any plane through the object, a mirror will appear to "swap" the object along that plane into its mirror image. Left/Right, top/bottom, front/back, even NNW/SSE.
It would also be easier to see with an object that had near-perfect vertical bilateral symmetry but no horizontal bilateral symmetry; the opposite of a human in a normal posture.
