The Standard Model differential cross section for ν¯e−e− elastic scattering vanishes exactly, at lowest order, for forward electrons and incident ν¯e energy close to the rest energy of the electron. This dynamical zero is not induced by a fundamental symmetry of the Lagrangian but by a destructive interference between the left- and right-handed chiral couplings of the electron in the charged and neutral current amplitudes. We show that lowest order analyses based on this favorable kinematic configuration are only mildly affected by the inclusion of the O(α) radiative corrections in the ν¯e−e− differential cross section, thus providing an excellent opportunity for the search of ``new physics''. In the light of these results, we discuss possible methods to improve the upper limits on the neutrino magnetic moment by selecting recoil electrons contained in a forward narrow cone. We conclude that, in spite of the obvious loss in statistics, one may have a better signal for small angular cones.