  We give a detailed study of the symplectic geometry of a family of integrable
systems obtained by coupling two angular momenta in a non trivial way. These
systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors
according to its value. For a certain range of values, the system is semitoric,
and we compute some of its symplectic invariants. Even though these invariants
have been known for almost a decade, this is to our knowledge the first example
of their computation in the case of a non-toric semitoric system on a compact
manifold (the only invariant of toric systems is the image of the momentum
map). In the second part of the paper we quantize this system, compute its
joint spectrum, and describe how to use this joint spectrum to recover
information about the symplectic invariants.
