  We study the spectral properties of a spin-boson Hamiltonian that depends on
two continuous parameters $0\leq\Lambda<\infty$ (interaction strength) and
$0\leq\alpha\leq\pi/2$ (integrability switch). In the classical limit this
system has two distinct integrable regimes, $\alpha=0$ and $\alpha=\pi/2$. For
each integrable regime we can express the quantum Hamiltonian as a function of
two action operators. Their eigenvalues (multiples of $\hbar$) are the natural
quantum numbers for the complete level spectrum. This functional dependence
cannot be extended into the nonintegrable regime $(0<\alpha<\pi/2)$. 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.
