The excess in electron recoil events reported recently by the XENON1T experiment may be interpreted as evidence for a sizable transition magnetic moment $mu_{ u_e u_mu}$ of Majorana neutrinos. We show the consistency of this scenario when a single component transition magnetic moment takes values $mu_{ u_e u_mu} in(1.65 - 3.42) times 10^{-11} mu_B$. Such a large value typically leads to unacceptably large neutrino masses. In this paper we show that new leptonic symmetries can solve this problem and demonstrate this with several examples. We first revive and then propose a simplified model based on $SU(2)_H$ horizontal symmetry. Owing to the difference in their Lorentz structures, in the $SU(2)_H$ symmetric limit, $m_ u$ vanishes while $mu_{ u_e u_mu}$ is nonzero. Our simplified model is based on an approximate $SU(2)_H$, which we also generalize to a three family $SU(3)_H$-symmetry. Collider and low energy tests of these models are analyzed. We have also analyzed implications of the XENON1T data for the Zee model and its extensions which naturally generate a large $mu_{ u_e u_mu}$ with suppressed $m_ u$ via a spin symmetry mechanism, but found that the induced $mu_{ u_e u_mu}$ is not large enough to explain recent data. Finally, we suggest a mechanism to evade stringent astrophysical limits on neutrino magnetic moments arising from stellar evolution by inducing a medium-dependent mass for the neutrino.
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