We analyze a family of models for a qubit interacting with a bosonic field. This family of models is very large and contains models where higher-order perturbations of field operators are added to the Hamiltonian. The Hamiltonian has a special symmetry, called spin-parity symmetry, which plays a central role in our analysis. Using this symmetry, we find the domain of self-adjointness and we decompose the Hamiltonian into two fiber operators each defined on Fock space. We then prove the Hunziker-van Winter-Zhislin (HVZ) theorem for the fiber operators, and we single out a particular fiber operator which has a ground state if and only if the full Hamiltonian has a ground state. From these results, we deduce a simple criterion for the existence of an excited state.