We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters 0 < or = Lambda < infinity (interaction strength) and 0 < or = alpha < or = 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 variant Planck's over 2pi ) 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.