Cosmological observations are promising ways to improve our understanding of neutrino mass properties. The upper bound on the sum of masses is given by the cosmic microwave background and large scale structure. These measurements are all parity-even, and potentially contaminated by unmodeled baryonic effects. In this paper we propose a novel parity-odd gravitational effect of neutrinos: A unique contribution to the directions of the angular momentum field of galaxies and halos. This observable is free of contamination in linear perturbation theory, and thus likely more cleanly separated from other nongravitational effects. A deep 21-cm survey to redshift 1 can potentially yield a 5σ significance on neutrino mass detection for a fiducial sum of neutrino masses of 0.05 eV.