  We present a theoretical formalism for scattering of the twisted neutrons by
nuclei in a kinematic regime where interference between Coulomb interaction and
the strong interaction is essential. Twisted neutrons have definite quantized
values of an angular momentum projection along the direction of propagation,
and we show that it results in novel observable effects for the scattering
cross section, spin asymmetries and polarization of the scattered neutrons. We
demonstrate that additional capabilities provided by beam's orbital angular
momentum enable new techniques for measuring both real and imaginary parts of
the scattering amplitude. Several possible observables are considered, for
which the targets may be either well-localized with respect to the spatial beam
profile, or the scattering occurs incoherently on nuclei in a bulk target. The
developed approach can be applied to other nuclear reactions with strongly
interacting twisted particles.
