Is it my mind's eye that is the problem or is this a "artifact" of the film? I would think it was my mind's eye as the medium of film is purely analog and not a digital representation that exists to conform to "what should be" and not in fact what is visually seen.
This happens when the wheel is observed as an image through a projector or with light shining on it that is periodically blinking. What you are seeing is essentially the additive effects of two periodic functions, the first is the period of the light flicker and the second is the period of the wheel rotation.
So, if the wheel is rotating at a frequency that is evenly divisible by the frequency of the light, then the spokes appear to be standing still. If it is rotating faster, then they are appear to be moving in one direction. If it is rotating slower, then it appears to be moving in the other.
Under household lighting, the light pulses are at 60 Hz. The frame rate of movie cameras is usually 24 Hz.
As John and Robin mentioned above, it relates to the superimposition of periodic functions. These functions can be an oscillating light source such as 60HZ AC lighting (power plants deliver electrical energy as oscillations to avoid the increased resistance presented by direct current), an oscillating image capture device such as a video camera, or the neuronal oscillations in your retina and primary visual cortex.
Let's start by going through the example where the "reverse spoke illusion" is caused by the interaction of the rotation and neural oscillation. We can then generalize this to all the forms of this illusion.
You perceive rotational motion when your eye "takes pictures" of a rotating object (we'll use a clockwise-rotating wheel with a single spoke for simplicity) as it moves. If the spokes in each picture are slightly clockwise of the spokes in the last picture, then higher order visual processing areas in your brain infer that the cause of this systematic difference in successive images is the physical rotation of the wheel. You experience this accordingly.
However, your brain only takes these pictures so fast. If the wheel is rotating slowly, the successive images are very similar, and it is easy to infer the clockwise rotation of the wheel from the series of small differences in spoke position. However, as the wheel spins faster and faster, the spokes travel farther and farther between pictures (the rate at which these pictures are taken is relatively fixed.) Because the wheel is rotating, it is a periodic function. Every time the wheel rotates once, it is in the same position it was in before. As the wheel continues to spin faster, the rate at which pictures are taken is exactly the same as the rotational period of the wheel. At this point, each picture will look exactly the same, because the wheel will be in the same position at each picture. The wheel will then appear to be fixed and not moving at all. At other speeds, each picture will be taken when the wheel spoke is slightly behind (counterclockwise of) where it was in the last picture. Since the brain infers motion by the relative distance between pictures, it will take this to mean that the wheel is traveling in the opposite direction!
This effect is multiplied by the number of spokes in the wheel. If the spokes are visually indistinguishable, then this period at which the wheel looks the same is the time it takes the wheel to rotate once divided by the number of spokes. A 5-spoked wheel will only need to spin 1/5th as fast for the same effect to occur.
You can now imagine how this phenomenon would present itself in other contexts with interacting oscillations. In the case of household lighting, the oscillations of the electric current effectively serve as the times at which pictures are taken (although this is a little more complicated because there are three sets of oscillations interacting, since your brain is still only taking pictures at a certain speed.) The same occurs when a digital or film camera takes pictures at a certain frequency.
Let me know if you could use some visual illustrations of the effect.