Another funny thing about mirrors is that they can be made to swap for real, i.e. a concave mirror.
Nope!
The reflection is only swapped when the focal point (or axis) is between the mirror and the observer. Thus, it is the
chosen path for the light that does the "swapping", and not the mirror: just put your eyeball close enough to the mirror, and the "swapping" always disappears.
However, a horizontally concave but vertically flat – somewhat like a cutout from a cylinder inner surface – mirror, when viewed from a suitable distance, does indeed reflect the image around the vertical axis, letting the observer see themselves somewhat like others see them. It only really works at a fixed distance, because further in or out the image aspect ratio will vary.
Cylindrical mirrors themselves have been used in anamorphic art: the mirror is placed on top of an image which is highly stretched, but when viewed through the mirror, the "correct" picture appears.