If we were dealing with a video camera, then this would likely be due to the stroboscopic effect. It is also called "the wagon wheel effect", and it's related to the Nyquist-Shannon sampling theorem.
The stroboscopic effect is a visual phenomenon caused by aliasing that occurs when continuous motion is represented by a series of short or instantaneous samples.
In the context of signal processing, "aliasing" refers to a situation where two inputs result in similar or identical outputs. Many fidget spinners have threefold rotational symmetry, which means that if you take it and rotate it one third away around (that is, 120 degrees), it looks the same: each three arms look the same, so once you rotate the spinner from one arm to the next, it looks the same as when you started. So if the camera frame rate is equal to the time it takes to rotate that much, the fidget spinner will look the same in each frame. This is called "aliasing" because a rotating fidget spinner is producing the same output as a stationary one.
So there will be some frequency that we can call the "fundamental frequency" that is indistinguishable from no rotation. Not only will this effect be produced at this rotational speed, but it will also be produced at any integer multiple: if the spinner is spinning at twice this speed, then two arms will go by, and you'll still end up with the spinner looking the same. Not only that, but any two frequencies that differ by an integer multiple of the fundamental frequency will appear the same. For instance, suppose you're spinning one spinner at 10% of the fundamental frequency, and another at 110% of it. Each frame, the first will get one tenth of the way to the next arm. The second spinner will get 10%% past the next arm. Since the arms look the same, this looks the same as getting 10% past the first arm.
This also means that going at 10% of the fundamental frequency in one direction looks the same as going 90% of the fundamental frequency in the other direction. If you spin it at 90% in the clockwise direction, then in the second frame it will be 10% away from the next arm, which looks the same as if it rotated 10% in the counterclockwise direction.
All of this applies to recording a fidget spinner, but what if you're directly looking at it? One way you could get a similar effect is if there's a flickering light source, and so your eyes are mostly seeing the fidget spinner at the peak of each flicker. The sun doesn't flicker, and the flickering of flames is generally not at a regular frequency, so you shouldn't see this effect with those light sources. However, AC power varies during its cycle (the standard in the US is 60 Hz), so with electricity-based light sources, there is regular flickering. This is especially pronounced with LEDs, less so with fluorescent, and even less with incandescent. If you can get all three light sources, you might want to see how strong the effect is which each. Laptop screens have a very regular refresh rate, so you should see a very strong strobe effect if you hold a fidget spinner up to a laptop screen.