A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poisson-commute and induce an $(\mathbb S^1 \times \mathbb R)$-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo $\&$ Vu Ngoc by means of five invariants.

Three of these five invariants are the so-called Taylor series invariant, the height invariant, and the twisting index. Roughly, the first one describes the behaviour of the system near the focus-focus singular fibre, the second one the position of the focus-focus value, and the third one compares the `distinguished' torus action given near each focus-focus singular fiber to the global toric `background action'.

Recently there has be made considerable progress in understanding and computing these invariants and, in this talk, we present the (results of the) finished and ongoing project with J. Alonso (Antwerp), H. Dullin (Sydney), and J. Palmer (Rutgers):

$\bullet$ Taylor series and twisting index for coupled spin oscillators.

$\bullet$ Taylor series, height invariant, and twisting index for coupled angular momenta.

$\bullet$ Putting the Taylor series and twisting index in relation with wellknown notions from classical dynamical systems like rotation number and rotation vector etc.

$\bullet$ Change of the Taylor series and twisting index when varying the parameters of these systems.