We present a general analysis of the effective potential for neutrino propagation in matter, assuming a generic set of Lorentz invariant non-derivative interactions. We find that in addition to the known vector and axial vector terms, in a polarized medium also tensor interactions can play an important role. We compute the effective potential arising from a tensor interaction. We show that the components of the tensor potential transverse to the direction of the neutrino propagation can induce a neutrino spin-flip, similar to the one induced by a transverse magnetic field.