  We consider two basic nuclear reactions: Radiative capture of neutrons by
protons, $n+p\to \gamma+~d$ and its time-reversed counterpart,
photodisintegration of the deuteron, $\gamma +d\to n+p$. In both of these cases
we assume that the incoming beam of neutrons or photons is "twisted" by having
an azimuthal phase dependence, {\it i.e.}, it carries an additional angular
momentum along its direction of propagation. Taking a low-energy limit of these
reactions, we derive relations between corresponding transition amplitudes and
cross sections with plane-wave beams and twisted beams. Implications for
experiments with twisted cold neutrons and photon beams are discussed.
