The Einstein-Cartan theory is a generalisation of General Relativity insofar as the condition that the metric affine connection is torsion-free is dropped. In other words, the space time is a Riemannian manifold together with the datum of a metric affine connection (which may differ from the Levi-Civita connection by a suitable contorsion tensor).
In this case, geodesics (paths that locally extremalise the length, and which are given by a variational principle) generally differ from auto-parallels. As far as I know, the trajectories of spin-less particles in Einstein-Cartan theory are usually assumed to be geodesics (rather than autoparallels) so they don't feel the difference between the given connection and the Levi-Civita connection. (By the way, is there a good reference for this statement?)
My question is, how classical particles with spin are supposed to behave? Will they also travel along geodesics with the only difference that their spin direction will evolve according to the contorsion tensor (viewed as a so(1,3)-valued one-form)?