  Given a quantum semitoric system composed of pseudodifferential operators,
Berezin-Toeplitz operators, or a combination of both, we obtain explicit
formulas for recovering, from the semiclassical asymptotics of the joint
spectrum, all symplectic invariants of the underlying classical semitoric
system. Our formulas are based on the possibility to obtain good quantum
numbers for joint eigenvalues from the bare data of the joint spectrum. In the
spectral region corresponding to regular values of the momentum map, the
algorithms developed by Dauge, Hall and the second author [27] produce such
labellings. In our proof, it was crucial to extend these algorithms to the
boundary of the spectrum, which led to the new notion of asymptotic
half-lattices, and to globalize the resulting labellings. Using the
construction given by Pelayo and the second author in [79], our results prove
that semitoric systems are completely spectrally determined in an algorithmic
way~: from the joint spectrum of a quantum semitoric system one can construct a
representative of the isomorphism class of the underlying classical semitoric
system. In particular, this recovers the uniqueness result obtained by Pelayo
and the authors in [62,61], and completes it with the explicit computation of
all invariants, including the twisting index. In the cases of the
spin-oscillator and the coupled angular momenta, we implement the algorithms
and illustrate numerically the computation of the invariants from the joint
spectrum.
