  Neutrinos propagating through matter may participate in forward coherent
neutral-current-like scattering arising from non-standard interactions as well
as from the Mikheyev-Smirnov-Wolfenstein matter potential $V_e$. We show that
at fixed long baselines through matter of constant density, the non-standard
interaction potential $\epsilon_{\mu\tau} V_e$ can contribute an additional
term to the oscillation phase whose sign differs for $\anumu$ versus $\numu$
propagation in matter. Its presence can cause different apparent $\Delta m^2$
to be erroneously inferred on the basis of oscillations in vacuum, with values
lying above (for $\anumu$) or below (for $\numu$) the actual $\Delta m^2_{32}$
for the case where $\epsilon_{\mu\tau}$ is predominantly real-valued and of
sign opposite to $\Delta m_{32}^2$. An NSI scenario invoking only
$\Re(\epsilon_{\mu\tau})$ is shown to be capable of accounting for a disparity
recently reported between oscillation survival for $\anumu$ and $\numu$ fluxes
measured at $735~\mathrm{km}$ by the MINOS experiment. Implications for mantle
traversal by atmospheric neutrinos are examined. The NSI matter potential with
non-maximal mixing could evade conventional atmospheric neutrino analyses which
do not distinguish $\numu$ from $\anumu$ on an event-by-event basis.
