My understanding of the spin of a particle is that it turns out, roughly speaking, your spinning tennis ball needs to keep track of a vector along the axis of rotation. This has something to do with the unit vectors in 2D Hilbert space having a correspondence to the group SU(2) (where remembering only the axis is the classical case of SO(3)). Somehow, if you were to take your quantum spinning tennis ball and, by some mechanism (magnetic fields?), slowly rotate its axis by 360 degrees, its spin becomes negative of what it originally was and it can destructively interfere with a tennis ball in the original state. Is this the sort of thing the pair of tennis balls is meant to deal with?
Something I really don't understand is how angular momentum is meant to work... I would have thought it's a Lie algebra thing, but SU(2) and SO(3) have isomorphic Lie algebras.
Disclosure: I don't know any quantum mechanics, just some math.