I will try my short explanation that I have been telling myself since the first time I heard this question. Imagine a mirror on the ceiling. From the same understanding that we are judging the image to be flipped left to right, we will now notice that the image is flipped top to bottom.

Explanation: Imagine a stick without width whose image is reflected in the mirror on the ceiling. If we superimpose the image on the stick, haven't we flipped top and bottom?