In this paper we advance into a generalized spinor classification, based on the so-called Lounesto’s classification. The program developed here is based on an existing freedom on the spinorial dual structures definition, which, in certain simple physical and mathematical limit, allows us to recover the usual Lounesto’s classification. The protocol to be accomplished here gives full consideration in the understanding of the underlying mathematical structure, in order to satisfy the quadratic algebraic relations known as Fierz–Pauli–Kofink identities, and also to provide physical observables. As we will see, such identities impose restrictions on the number of possible spinorial classes allowed in the classification. We also expose a subsidiary mathematical device—a slight modification on the Clifford algebra basis—which ensures real spinorial densities and holds the Fierz–Pauli–Kofink quadratic relations.