Not OP, but: it works similarly to looking at the system in a mirror. The clock's hands turn, well, clockwise, but if you look at the mirror their movement is anticlockwise. Importantly, if you look at that mirror in another mirror, it will be clockwise again. Add yet another mirror and it's anticlockwise.

With a single mirror at position x=0 (and YZ plane), you invert "x" position, so (1, 1, 1) becomes (-1, 1, 1). "Inverting" the spatial coordinates ((x,y,z) -> (-x, -y, -z)) is effectively the same as looking at system through 3 mirrors, located at x = 0 (YZ plane), y = 0 (XZ plane) and z = 0 (XY plane), but that is a bit hard to visualize/arrange in practice so usually you would think of it as an equivalent operation of a point reflection around (0, 0, 0). You are right that the point is arbitrary: the important thing is, among others, that clockwise movement becomes anticlockwise.