We consider the Riemann-Cartan geometry as a basis for the Einstein-Sciama-Kibble theory coupled to spinor fields: we focus on f(R) and conformal gravities, regarding the flag-dipole spinor fields, type-(4) spinor fields under the Lounesto classification. We study such theories in specific cases given, for instance, by cosmological scenarios: we find that in such background the Dirac equation admits solutions that are not Dirac spinor fields, but in fact the aforementioned flag-dipoles ones. These solutions are important from a theoretical perspective, as they evince that spinor fields are not necessarily determined by their dynamics, but also a discussion on their structural (algebraic) properties must be carried off. Furthermore, the phenomenological point of view is shown to be also relevant, since for isotropic Universes they circumvent the question whether spinor fields do undergo the Cosmological Principle.

We study Dirac spinors in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity. We find four types of results: the resulting dynamic behavior of the universe depends on the particular choice of function f(R); some f(R) models do not isotropize and have no Einstein limit, so that they have no physical significance, whereas for other f(R) models isotropization and Einsteinization occur, and so they are physically acceptable, suggesting that phenomenological arguments may select f(R) models that are physically meaningful; the singularity problem can be avoided, due to the presence of torsion; the general conservation laws holding for f(R)-gravity with torsion ensure the preservation of the Hamiltonian constraint, so proving that the initial value problem is well-formulated for these models.