We investigate new symmetries which decouple the mass from the magnetic moment of neutrinos and their theoretical and phenomenological implications are discussed. Our proposed model is based on $SU(2)_H$ horizontal symmetry that can generate large neutrino transition magnetic moment without inducing unacceptably large neutrino masses. In the $SU(2)_H$ symmetric limit, the transition magnetic moment is nonzero, while the neutrino mass vanishes. The simplification we suggest is based on the symmetry being approximate, which we also generalize to a three-family $SU(3)_H$-symmetry. We have also investigated a spin symmetry mechanism that can generate large $\mu_\nu$ while keeping $m_\nu$ small. This talk is based on results obtained with K.S. Babu, and Manfred Lindner and presented in hep-ph 2007.04291~\cite{Babu:2020ivd}.