  The excess in electron recoil events reported recently by the XENON1T
experiment may be interpreted as evidence for a sizable transition magnetic
moment $\mu_{\nu_e\nu_\mu}$ of Majorana neutrinos. We show the consistency of
this scenario when a single component transition magnetic moment takes values
$\mu_{\nu_e\nu_\mu} \in(1.65 - 3.42) \times 10^{-11} \mu_B$. 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, $m_\nu$ vanishes while
$\mu_{\nu_e\nu_\mu}$ 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 $\mu_{\nu_e\nu_\mu}$ with
suppressed $m_\nu$ via a spin symmetry mechanism, but found that the induced
$\mu_{\nu_e\nu_\mu}$ 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.
