  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.
