We present the principles of variable pure polarization rotators based on the accumulation of a geometric phase that is obtained by coiling a beam of light via discrete reflections. They consist of two pseudo rotators in series. By changing the angle between the pseudo rotators the geometric phase acquired by the light is varied. We describe in detail two important examples: the variable-angle Porro rotator, which uses four linear-polarization preserving reflectors to work at a predetermined wavelength range, and the variable compensating phase-shift rotator, which uses eight reflections to compensate the overall s- and p-polarization phase shifts, and thus is achromatic.