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We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters 0≤Λ<∞ (interaction strength) and 0≤α≤π/2 (integrability switch). In the classical limit, this system has two distinct integrable regimes, α=0 and α=π/2. For each integrable regime we can express the quantum Hamiltonian as a function of two action operators. Their eigenvalues (multiples of ℏ) are the natural quantum numbers for the complete level spectrum. This functional dependence cannot be extended into the nonintegrable regime (0<α<π/2). Here level crossings are prohibited and the level spectrum is naturally described by a single (energy sorting) quantum number. In consequence, the tracking of individual eigenstates along closed paths through both regimes leads to conflicting assignments of quantum numbers. This effect is a useful and reliable indicator of quantum chaos—a diagnostic tool that is independent of any level-statistical analysis.