>The words “up” and “down” are absolute, but the words “left” and “right” are relative to you. If we instead of “left” and “right” use the words “south arm” and “north arm”, then the mirror suddenly does not “flip” anything.

No dude. The mirror flips left and right Relative to you. The mirror ignores up and down relative to you. The paradox is real when thought of in these terms.

Truth be told the flip is neither left or right nor up and down. actually It is actually inside out on the z axis: towards you. Translational coordinates along the xy plane must be preserved and this is what causes the perceived left right flip.

The actuality is nothing was flipped along x or y, it was kept the same.

That's what trips everyone up.