  Experimental and theoretical investigations of excitons in cuprous oxide have
revealed a significant fine-structure splitting of the excitonic Rydberg states
caused by a strong impact of the valence band structure. We provide a
semiclassical interpretation of that splitting by investigating the classical
dynamics of the excitonic electron-hole pair beyond the hydrogen-like model.
Considering the slow motion of Rydberg excitons in coordinate space compared to
the fast dynamics of quasispin and hole spin we use an adiabatic approach and
energy surfaces in momentum space for the computation of the exciton dynamics.
We observe quasi-periodic motion on near-integrable tori. Semiclassical torus
quantization yields the energy regions of the fine-structure splitting of
$n$-manifolds in agreement with quantum mechanical computations.
