There's an experiment that transfers that angular momentum all the way up to macroscopic levels. By magnetizing a cylinder of iron, all the spins start pointing in the same direction. By conservation of angular momentum, the cylinder itself has to start spinning in the opposite direction. I'm very fond of this experiment, because it magnifies a strange quantum phenomenon to the classical level.
Spin being an intrinsically quantum mechanical concept, I'm afraid the microscopic mechanism by which that transfer occurs will only be explainable in a quantum mechanical context. Here it will appear as a term in the Hamiltonian coupling the spin of an electron to its motion in a potential.
Do you know an example of a process that moves angular momentum from one kind of spin to the other?