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.
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.
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.)