  We examine a phenomenon recently predicted by numerical simulations of
supernova neutrino flavor evolution: the swapping of supernova $\nu_e$ and
$\nu_{\mu,\tau}$ energy spectra below (above) energy $\EC$ for the normal
(inverted) neutrino mass hierarchy. We present the results of large-scale
numerical calculations which show that in the normal neutrino mass hierarchy
case, $\EC$ decreases as the assumed $\nu_e\rightleftharpoons\nu_{\mu,\tau}$
effective $2\times 2$ vacuum mixing angle ($\simeq \theta_{1 3}$) is decreased.
However, these calculations also indicate that $\EC$ is essentially independent
of the vacuum mixing angle in the inverted neutrino mass hierarchy case. With a
good neutrino signal from a future Galactic supernova, the above results could
be used to determine the neutrino mass hierarchy even if $\theta_{13}$ is too
small to be detected in terrestrial neutrino oscillation experiments.
