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 $epsilon_{mutau} V_e$ can contribute an additional term to the oscillation phase whose sign differs for $anumu$ versus $ umu$ 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 $ umu$) the actual $Delta m^2_{32}$ for the case where $epsilon_{mutau}$ is predominantly real-valued and of sign opposite to $Delta m_{32}^2$. An NSI scenario invoking only $Re(epsilon_{mutau})$ is shown to be capable of accounting for a disparity recently reported between oscillation survival for $anumu$ and $ umu$ 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 $ umu$ from $anumu$ on an event-by-event basis.
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