Thanks to Schwinger [Phys. Rev. 73, 407 (1948)PHRVAO0031-899X10.1103/PhysRev.73.407], the process of elastic scattering of neutrons by nuclei is known to depend on the interference between a nuclear amplitude and an electromagnetic one for small scattering angles, resulting in spin asymmetries of a cross section or in polarization of the scattered neutrons. Although this interference depends on the neutron's transverse polarization and on an imaginary part of the nuclear amplitude, this conclusion holds only for the incident plane-wave neutrons with a definite momentum. Here, we show that this scattering is altered when the twisted neutrons, recently obtained experimentally, are used instead - that is, neutrons with an orbital angular momentum. For bulk targets, the angular distributions of the scattered neutrons get modified, whereas scattering of a superposition of states with the different angular momenta also reveals dependence on the longitudinal polarization. For well-localized targets, the observables develop dependence on the neutron's helicity and on a real part of the nuclear amplitude, providing full access to its phase already in the Born approximation. We argue that the corresponding spin asymmetries are measurable at existing neutron facilities. Thus, scattering of the twisted neutrons by nuclei can provide means for quantum tomography of the neutron states and become a useful tool for hadronic studies, low-energy nuclear physics, tests of fundamental symmetries, and neutron optics.