The origin and the properties of the Hermitian displacement-parity operator as well as the consequent implications are investigated. This operator is defined as product of the displacement and parity operators, both introduced by Glauber in the sixties in a series of his seminal papers on the quantum treatment of optical coherence. The link between the operator under scrutiny and the existence of an unexpected invariance property of the Hamiltonian model that describes a driven quantum harmonic oscillator reveals its origin and explains its physical relevance. This knowledge paves the way for an extensive study of its physical properties and the construction of new mathematical identities. Intriguing possible developments of this investigation are briefly highlighted.