How come a plane mirror does lateral inversion (left to right), not top to bottom?
Actually the mirror does not do any kind of lateral inversion!
The first line above shows an object and what it appears like in a mirror.
The second line shows the conventional way our brain interprets the image: We imagine that the object has moved behind the mirror by going around it as the arrow shows. When we imagine this, to make the object coincide with what we see, we need to flip left and right. In other words, we imagine that the mirror has flipped left and right.
The third line shows another possible interpretation. We imagine that the object has gone behind the mirror by flipping OVER the mirror. When we imagine this, to make the object coincide with what we see, we need to flip TOP and BOTTOM! In other words, we would imagine that the mirror has flipped top and bottom!
The mirror has neither flipped left and right nor has it flipped top and bottom. The right side of the image is on the right side, the left side on the left, the top on the top and the right on the right!
Can we have more complex imagination? If we imagine that the object has moved behind the mirror by following a line making an arbitrary angle with the horizontal, it would have to be flipped along the perpendicular line in order to make it coincide with the image. Why then do we always stick to a single interpretation? It is more natural for the brain to imagine that we have walked to the other side than to imagine that we have jumped over the mirror!

Why does the brain imagine that the object has crossed the mirror? That is because of what the mirror actually does: It reverses front and back! A part of an object that is closer to the mirror images as a point that is closer to the mirror and vice versa. In order to comprehend this, the brain imagines that there is another object behind the mirror, but facing in the opposite way. How did the second object get there? Obviously by going behind the mirror!