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Can anything spin in two directions at once, separately? If so, how? For instance, could an object spin sideways and up and down at the same time, separately?

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The notion (and word) spin for subatomic particles is NOT the same as spin for objects such as a football. It's a purely mathematical idea, that unfortunately uses the same word as classical/normal objects. Elementary objects are, as far as is known, pointlike, so convential spinning does not apply to them.

A classical object, such as a football, can have it's spin resolved into perpendicular components, but I assume you mean two simultanously opposite directions, and no that's not possible.

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