  We show that chiral symmetry and gauge invariance enforce relations between
the short-distance physics that occurs in a number of electroweak and pionic
reactions on light nuclei. Within chiral perturbation theory this is manifested
via the appearance of the same axial isovector two-body contact term in pi- d
-> n n gamma, p-wave pion production in NN collisions, tritium beta decay, pp
fusion, nu d scattering, and the hep reaction. Using a Gamow-Teller matrix
element obtained from calculations of pp fusion as input we compute the neutron
spectrum obtained in pi- d -> n n gamma. With the short-distance physics in
this process controlled from p p -> d e+ nu_e the theoretical uncertainty in
the nn scattering length extracted from pi- d -> n n gamma is reduced by a
factor larger than three, to <~0.05 fm.
