Why does slight facial asymmetry look obvious in a photo when nobody notices it in person?

Three separate things are happening and they compound.

A photo is one instant, frozen. A real face is constantly moving, and your brain builds an average from thousands of micro-expressions. A photo hands you one of those samples at random. If the shutter caught the moment one side of the smile was ahead of the other, and it usually is, because faces are not synchronised - you get a lopsidedness that never existed as a sustained state.

A photo is flat. In person, depth cues from binocular vision and your own movement tell you a cheek is turned slightly away. Flattened into two dimensions, that turn reads as a smaller eye or a shorter side rather than a rotation.

A photo can be stared at. You never study a face for thirty seconds in life. Any feature survives a glance and very few survive scrutiny.

So it is not that the camera revealed something. It sampled, flattened and then invited inspection.

The mirror thing is real and it is probably most of why the subject reacts more strongly than you do.

You have seen your own face mostly reversed, tens of thousands of times, and that reversed version is the one that feels like you. A photograph shows the unreversed face, so every asymmetry sits on the opposite side from where you know it. It is unfamiliar in a way that is hard to name, which is why people say a photo "does not look like me" rather than "my nose is on the wrong side".

The practical test, which is fun to do with someone who hates their photos: show them a mirrored copy. Most people prefer it instantly and cannot say why. Their friends usually prefer the unmirrored one, because that is the face they know.

This is why front-camera apps that save the mirrored preview feel flattering and the unmirrored file feels wrong. Same photo.

Things that actually help while shooting, roughly in order of effect:

Get further away and zoom in. Close and wide is the single biggest cause of faces looking odd: the nearer features enlarge relative to the further ones, so a head turned a few degrees becomes dramatically uneven. Two or three metres back with a longer lens flattens that out. This one change fixes more "bad photos of me" than everything else combined.

Shoot slightly off a dead-on angle, and pick a side. Straight on maximises the left-right comparison, because the two halves sit side by side for the eye to check. A small turn gives the brain a rotation to explain any difference with.

Soft, frontal-ish light. Hard light from one side carves one half of the face differently from the other and doubles any asymmetry. A window at an angle, or open shade, is forgiving.

Take more frames, and take them while they are talking. You are sampling. Sample more.

The non-technical half, which matters more than any of the above for people who dislike being photographed: do not show them the back of the camera between shots.

The moment someone sees a frame they dislike, they tense up, and a tense face is more asymmetric than a relaxed one - the muscles on the dominant side pull harder. So you get a feedback loop where reviewing the photo produces worse photos.

Shoot, keep talking, do not review until later, and choose for them rather than handing over the whole set. It sounds like a trick. It is mostly just not interrupting.