Why do left and right flip but top and bottom do not in a mirror?
After all, the plane of mirroring is neutral to all directions parallel to the plane.

The answer of course is in the form of an axis of rotation preference. Rotation about the bodyline axis (left right flip) and rotation about waist axis (top bottom flip) are not equivalent. But let’s first be clear that the mirror does treat them equivalently in terms of optics, so that’s not where the non-equivalence comes from. In fact, it is futile to think too much about the mirror.

Instead, let’s think about the person. When you physically turn around to face the other way, you make a choice to rotate around one particular axis, and not the other. That’s a very obvious breaking of equivalence. When you look at somebody else or at your image in the mirror, you are likewise choosing a preferred axis — except in your mind. You make a mental rotation and that’s where the equivalence is broken. That’s why left and right flip but top and bottom do not, because your mind chooses to do the left-right flip and not the top-bottom flip.

But why does the mind choose one rotation over the other? Because one can be done, and the other can’t. Left and right in the body plan are symmetric, so the image seen in a mirror makes “sense” as a rotated image along the bodyline axis. It would not make “sense” as a rotated image along the waist axis, because head and feet are not symmetric.

On a related but different matter, if you look into a lake, you see a top-bottom flip. That’s perfectly normal because the plane of mirroring in this case is itself preferential. In this case, it is a purely optical equivalence breaking. So far so good. But you are somehow aware that the image in the lake is abnormal (i.e. upside down). Now if you face a wall mirror sideways, you have likewise a left-right optically flipped image. But nothing seems abnormal there. What is it about upside down that is abnormal? Well, it has to be due to the external reference of updown-ness known as gravity. In fact, if you stayed in the space shuttle long enough, I bet seeing an upside down person or image would no longer seem at all weird.