  We explain anomalies currently present in various data samples used for the
measurement of the anomalous magnetic moment of electron ($a_e$) and muon
($a_\mu$) in terms of an Aligned 2-Higgs Doublet Model with right-handed
neutrinos. The explanation is driven by one and two-loop topologies wherein a
very light CP-odd neutral Higgs state ($A$) contributes significantly to
$a_\mu$ but negligibly to $a_e$, so as to revert the sign of the new physics
corrections in the former case with respect to the latter, wherein the dominant
contribution is due to a charged Higgs boson ($H^\pm$) and heavy neutrinos with
mass at the electroweak scale. For the region of parameter space of our new
physics model which explains the aforementioned anomalies we also predict an
almost background-free smoking-gun signature of it, consisting of $H^\pm A$
production followed by Higgs boson decays yielding multi-$\tau$ final states,
which can be pursued at the Large Hadron Collider.
