Yes, it's true "put a mirror on the floor... and you will see yourself feet upwards" (Book review, October 1). But exactly why? A plane mirror inverts space in whichever direction is perpendicular to its surface. If Alice faces north and looks into a vertical mirror with its surface east and west, her east side hand has an east side reflection and her west side hand has a west side reflection - she can put her palms up to touch at the surface. Her reflection's up is up and its down is her down.
But her north-facing nose has a south-facing reflection; the north-south axis perpendicular to the reflecting surface has been reversed.
Alice wonders why she cannot rotate herself to match her reflection: it's because the light rays coming away from the plane mirror produce only an inversion (not a rotation), ie a reversal of the direction perpendicular to the mirror. A mirror on the horizontal floor inverts the vertical axis so Alice's reflection is upside down. Her reflected east is east and her north-facing nose has a north-facing upside down nose reflection.