Why do mirrors flip horizontally and not vertically?
tl;dr We "see" horizontal rotation because it's common. When we want to face in the opposite direction we rotate something horizontally, not vertically. And the object in the mirror looks like it's facing us.
The reason it looks like mirrors rotate us horizontally is a combination of two things.
First, our eyes interpret the things we see as physical objects. So when we look at a mirror, we interpret what we see as objects which can exist in the real world.
And second, gravity causes objects to have an obvious up side and down side, and the two cannot be practically switched. So when we want a car to face opposite the direction it's facing now, we rotate it horizontally. And the same goes for when we want to face someone. And if it's something we can't move like a mountain, we will go to its other side and rotate ourselves horizontally. We won't jump over the mountain landing on our head on its other side. This has the effect of us paying much more attention to an object's up-down shape than to its right-left shape. In fact, it's possible that we don't even remember which side is the correct side for things, left or right. A rotated asymmetrical puppet would still look normal in the mirror despite it's right and left being switched, while the same puppet placed on its head, won't.
Taking the two combined, we more readily interpret what we see in the mirror as a horizontally rotated person and ignore their shirt pocket's placement, than interpret it as a person that has rotated vertically (and then need to ignore the fact that their head is pointing upwards, which wouldn't be the case then).
This is then assisted by the fact that many things, including people, are mostly symmetrical horizontally, so the person we see in the mirror is quite similar to what we would look like if we would rotate horizontally.
To see that a mirror does not, in fact, show us rotated horizontally - stand with your right hand stretched out to your right pointing at an object there. Now turn half a turn. Now your hand is pointing away from that object. Now stand as you were before you turned and instead of turning, look at a mirror. You'll see that your reflection is pointing at the object.