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A discrete parity $\mathbb{Z}_2$ symmetry of a two parameter extension of thequantum Rabi model which smoothly interpolates between the latter and theJaynes-Cummings model, and of the two-photon and the two-mode quantum Rabimodels enables their diagonalization in the spin subspace. A more generalstatement is that the respective sets of $2\times 2$ hermitian operators of theFulton-Gouterman type and those diagonal in the spin subspace are unitaryequivalent. The diagonalized representation makes it transparent that anyquestion about integrability and solvability can be addressed only at the levelof ordinary differential operators of Dunkl type. Braak's definition ofintegrability is shown (i) to contradict earlier numerical studies and (ii) toimply that any physically reasonable differential operator of Fulton-Goutermantype is integrable.

Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type