  The cosmological model in which 20% of the dark matter is shared by two
nearly equal mass neutrinos fits the structure of the universe on all scales.
This has been motivated a $\nu_\mu$-$\nu_{\tau}$ oscillation explanation of the
deficit of atmospheric muon neutrinos. If the observed ratio of atmospheric
$nu_\mu$ to $\nu_e$ has an alternative explanation, the cosmological model can
be retained if the deficit of solar neutrinos is explained by
$\nu_e$-$\nu_{\tau}$ oscillation. In this case an inverted mass hierarchy is
required with $m_{\nu_{\mu}}\ll m_{\nu_e} \simeq m_{\nu_\tau}\approx 2.4$ eV.
We show that if there exists an $L_e- L_{\tau}$ symmetry in nature, both the
near mass degeneracy of \nue\ and \nut\ as well as the consistency of the above
values for neutrino masses with the negative results for neutrinoless double
beta decay search experiments are easily understood. We show that this symmetry
implemented in the context of a high-scale left-right symmetric theory with the
see-saw mechanism can lead to a simple theoretical understanding of the desired
form of the mass matrix.
