  This article introduces the notion of good labellings for asymptotic lattices
in order to study joint spectra of quantum integrable systems from the point of
view of inverse spectral theory. As an application, we consider a new spectral
quantity for a quantum integrable system, the quantum rotation number. In the
case of two degrees of freedom, we obtain a constructive algorithm for the
detection of appropriate labellings for joint eigenvalues, which we use to
prove that, in the semiclassical limit, the quantum rotation number can be
calculated on a joint spectrum in a robust way, and converges to the well-known
classical rotation number. The general results are applied to the semitoric
case where formulas become particularly natural.
