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Sorry for the high number of physics posts, but I don’t want to let the crowd over at 3v1l 5c13nc3 down. This problem confused André and I one day as undergraduates back at MUN, so I particularly enjoy it.
Why does a flat, plane mirror seem to reverse your image left/right but not up/down?

Image credit: Donald Simanek
The case of the plane and the conveyor belt Textpattern update
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.
Mirror reflection is inversion of one coordinate (the mirror axis normal to the plane of reflection). That inverts the overall sign of the coordinate system via the vector triple product. Reflected chiral objects reverse handedness.
Parity inverts all coordinates without spatial bias. Graphically and with symmetry element explanation plus literature citations,
http://www.mazepath.com/uncleal/eotvos.htm#b3
In other, less technical terms, it doesn’t reflect left and right, it reflects “in” and “out”, where “in” means pointing towards the mirror and “out” means pointing away from the mirror.