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The simplest answer to this apparent contradiction: When you look in the mirror, your right ear appears on the right-hand side of the mirror, and your left ear on the left. So surely you were not expecting the top of your head to appear at the bottom of the mirror and your neck at the top? Of course you weren't, by the same argument.
A deeper answer (but still basically very simple): Any object that does not have a plane of symmetry is said to be chiral (in chemistry the term and the concept is often applied to molecules, and is especially important in biochemistry and pharmacology). When you view a chiral object in a mirror, the image that you see has the chirality reversed. Thus, for example, you look like someone with hair parted on the left, whereas yours is actually parted on the right. This is called "lateral inversion". It is an unfortunate and misleading term, as it seems to suggest a sideways inversion, i.e. that left and right are interchanged in the image that we see, which certainly does not happen in the sense of shifting from one side to the other (see above). The reversal of chirality is the complete story - there's really nothing more to be said about it. No question about the image being turned upside-down should arise.
Comment: Here's a fairly simple explanation, but it's "deep" too and it's correct.
1) Lateral inversion occurs with plane (flat) mirrors.
2) The mirror only SEEMS to reverse right and left. So that gets rid of the problem in the question, but it raises the question of why it SEEMS to reverse
right and left.
3) The mirror actually just reverses points on the object in the direction that's
perpendicular to the mirror's surface. That's ALL it does.
4) However, the mirror image is often perceived ("seen") as being left-right reversed, with front-back being "seen" as unchanged.
That's the bit that's hard to explain. It's to do with the psychology of visual perception. If you look at yourself in a mirror it is really hard to try to "see" yourself as reversed front to back.
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