Massive neutrinos have nonzero magnetic moments. It is known that the neutrino spin oscillations can be induced by the neutrino magnetic moment interaction with the transversal magnetic field. We show that the effect of neutrino spin oscillations can be engendered in the absence of the neutrino magnetic moment and without of the transversal magnetic field. From a regions derivation of the effective neutrino evolution Hamiltonian in moving matter it follows that spin oscillations can be engendered by the interaction with the transversal current of matter. The obtained general expression for the neutrino effective evolution Hamiltonian can be used for investigations of various types of neutrino spin oscillations in the transversal matter currents considered in the neutrino mass and flavour basis.