  Precision measurements of neutrino oscillation parameters have provided a
tremendous boost to the search for sub-leading effects due to several beyond
the Standard Model scenarios in neutrino oscillation experiments. Among these,
two of the well-studied scenarios are Lorentz violation (LV) and non-standard
interactions (NSI), both of which can affect neutrino oscillations
significantly. We point out that, at a long-baseline experiment where the
neutrino oscillation probabilities can be well-approximated by using the
line-averaged constant matter density, the effects of these two scenarios can
mimic each other. This would allow the limits obtained at such an experiment on
one of the above scenarios to be directly translated to the limits on the other
scenario. However, for the same reason, it would be difficult to distinguish
between LV and NSI at a long-baseline experiment. We show that the observations
of atmospheric neutrinos, which travel a wide range of baselines and may
encounter sharp density changes at the core-mantle boundary, can break this
degeneracy. We observe that identifying neutrinos and antineutrinos separately,
as can be done at INO-ICAL, can enhance the capability of atmospheric neutrino
experiments to discriminate between these two new-physics scenarios.
