We consider two basic nuclear reactions: Radiative capture of neutrons by protons, $n+pto gamma+~d$ and its time-reversed counterpart, photodisintegration of the deuteron, $gamma +dto n+p$. 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.
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