In an unconventional realization of left-right symmetry, the particle corresponding to the left-handed neutrino nu_L (with SU(2)_L interactions) in the right-handed sector, call it n_R (with SU(2)_R interactions), is not its Dirac mass partner, but a different particle which may be a dark-matter candidate. In parallel to leptogenesis in the SU(2)_L sector, asymmetric production of n_R may occur in the SU(2)_R sector. This mechanism is especially suited for n_R mass of order 1 to 10 keV, i.e. warm dark matter, which is a possible new paradigm for explaining the structure of the Universe at all scales.