We show how a list of symmetry principles can be used to generate a cl
ass of models in which the neutrino masses are suppressed while relati
vely large magnetic moments are allowed. The simplest example is a mod
el for neutrino mass proposed by Zee some time ago. We show how the mo
del can be improved based on the symmetry principle and we also demons
trate other models in this class which avoid some of the weakness in t
he simplest model.