Is spin/rotation a unique and lonely euclidean function?

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Is there something more complex than a 2 dimensional "spin"? I wouldn't know how to describe it. We can spin 3 dimensional objects in 3 ways, but we can only spin them on an axis. All spins happen on an axis. Is it possible for a higher-dimensional spin function to exist? Would it "rotate" around a plane? Is this at all related to the Casimir operator?