Right, the mirror doesn't actually flip. It's just we think about turning around to try to "match" what we see.

And humans are more left-right symmetric than up-down symettric.

If we try to do a headstand to match the image in the mirror then it actually "mirrors" it top/down instead of left/right.

Another example: looking at writing in the mirror, we've actually turned it around to point at the mirror. If we flip it over to point it at the mirror instead then again it'll appear upside down in the mirror instead of left/right flipped.

Edit: I should have read the article, because indeed this is exactly what it explains.