  Quantum semitoric systems form a large class of quantum Hamiltonian
integrable systems with circular symmetry which has received great attention in
the past decade. They include systems of high interest to physicists and
mathematicians such as the Jaynes\--Cummings model (1963), which describes a
two-level atom interacting with a quantized mode of an optical cavity, and more
generally the so-called systems of Jaynes\--Cummings type. In this paper we
consider the joint spectrum of a pair of commuting semiclassical operators
forming a quantum integrable system of Jaynes\--Cummings type. We prove,
assuming the Bohr\--Sommerfeld rules hold, that if the joint spectrum of two of
these systems coincide up to $\mathcal{O}(\hbar^2)$, then the systems are
isomorphic.
