We consider a nonstandard interaction between the neutrino and a hypothetical massive scalar or pseudoscalar. In the ultrarelativistic limit, the new effective interaction between the neutrinos vanishes if neutrinos are Dirac fermions but not if they are Majorana fermions. Using a multi-angle ``neutrino bulb model'' we calculate the flavor transformation above the neutrinosphere in a core collapse supernova and find that the addition of the nonstandard neutrino self-interaction (NSSI) to the ordinary V-A self-interaction between neutrinos is capable of dramatically altering the collective oscillations. If our understanding of the neutrino signal from a Galactic supernova is sufficiently well understood, supernova neutrinos provide complimentary constraints on scalar/pseudoscalar interactions of neutrinos.