Future large neutrino detectors, such as JUNO, Hyper-Kamiokande and DUNE, will open a new era in studies of supernova and presupernova neutrinos, which can give important information for the explosion mechanism of collapse-driven supernovae. This work is dedicated to the neutrino evolution in the astrophysical environment, taking into account the collective neutrino oscillations between different helical states engendered by interaction of the neutrino magnetic moment with a magnetic field, as well as the interaction with matter in the case when there is a transversal matter current or matter polarization. By using the numerical simulation method, we show that distinct spin-flavor spectral splits do exist in the supernova neutrino flavor evolution when there is non-trivial transversally moving matter.