  Thanks to J.~Schwinger, 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.
While this interference depends on the neutron's {\it transverse} polarization
and on {\it 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, while 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 a dependence
on the neutron's {\it helicity} and on {\it 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.
