Collier N. Smith is a technical writer at Boulder Laboratories of the National Institute of Standards and Technology. Here is his reply.
This phenomenon is not common in direct vision, but often seen in movies and television. Film and TV actually consist of series of still photographs shown in rapid succession to fool the eye into seeing motion. The individual pictures do not actually move. Instead, the position of a moving object shifts with respect to the background in each successive still frame; when viewed rapidly, one gets an impression of smooth motion.
The backward motion illusion occurs when the speed of rotation is such that--in the interval between each frame--a new blade moves nearly into the position occupied by a blade in the previous frame. If the timing is precise, it looks like the propeller did not turn at all.
But if the blade interval doesn't exactly match the camera interval (or a multiple of it), then the propeller will seem to turn slowly forward or backward. When the next blade is a little slow in reaching the previous blade's position, the rotation appears to be backward; when the next blade arrives a fraction early, the visual impression is that the blades are turning forward.
The same phenomenon can be observed by adjusting the number of light flashes from a stroboscope. Under certain circumstances, objects illuminated by some kinds of fluorescent lights will appear to rotate backwards to the naked eye. The effect occurs due to the 60 cycle per second flickering of the light, which is normally too rapid for the eye to register.
When the blade interval and frame interval are quite different from one another, the blades become blurred and the phenomenon disappears.