  The coupled angular momenta are a family of completely integrable systems
that depend on three parameters and have a compact phase space. They correspond
to the classical version of the coupling of two quantum angular momenta and
they constitute one of the fundamental examples of so-called semitoric systems.
Pelayo & Vu Ngoc have given a classification of semitoric systems in terms of
five symplectic invariants. Three of these invariants have already been
partially calculated in the literature for a certain parameter range, together
with the linear terms of the so-called Taylor series invariant for a fixed
choice of parameter values.
  In the present paper we complete the classification by calculating the
polygon invariant, the height invariant, the twisting-index invariant, and the
higher-order terms of the Taylor series invariant for the whole family of
systems. We also analyse the explicit dependence of the coefficients of the
Taylor series with respect to the three parameters of the system, in particular
near the Hopf bifurcation where the focus-focus point becomes degenerate.
