  We investigate cosmological scenarios with spin-gravity coupling. In
particular, due to the spin of the baryonic and dark matter particles and its
coupling to gravity, they probe an effective spin-dependent metric, which can
be calculated semi-classically in the Mathisson-Papapetrou-Tulczyjew-Dixon
formalism. Hence, the usual field equations give rise to modified Friedmann
equations, in which the extra terms can be identified as an effective
dark-energy sector. Additionally, we obtain an effective interaction between
the matter and dark-energy sectors. In the case where the spin-gravity coupling
switches off, we recover standard $\Lambda$CDM cosmology. We perform a
dynamical system analysis and we find a matter-dominated point that can
describe the matter era, and a stable late-time solution corresponding to
acceleration and dark-energy domination. For small values of the spin coupling
parameter, deviations from $\Lambda$CDM concordance scenario are small, however
for larger values they can be brought to the desired amount, leading to
different dark-energy equation-of-state parameter behavior, as well as to
different transition redshift from acceleration to deceleration. Finally, we
confront the model predictions with Hubble function data.
