  Neutrinos are the fundamental particles, blind to all kind of interactions
except the weak and gravitational. Hence, they can propagate very long
distances without any deviation. This characteristic property can thus provide
an ideal platform to investigate Planck suppressed physics through their long
distance propagation. In this work, we intend to investigate CPT violation
through Lorentz invariance violation (LIV) in the long-baseline accelerator
based neutrino experiments. Considering the simplest four-dimensional Lorentz
violating parameters, for the first time, we obtain the sensitivity limits on
the LIV parameters from the currently running long-baseline experiments T2K and
NO$\nu$A. In addition to this, we show their effects on mass hierarchy and CP
violation sensitivities by considering NO$\nu$A as a case study. We find that
the sensitivity limits on LIV parameters obtained from T2K are much weaker than
that of NO$\nu$A and the synergy of T2K and NO$\nu$A can improve these
sensitivities. All these limits are slightly weaker ($2 \sigma$ level) compared
to the values extracted from Super-Kamiokande experiment with atmospheric
neutrinos. Moreover, we observe that the mass hierarchy and CPV sensitivities
are either enhanced or deteriorated significantly in the presence of LIV as
these sensitivities crucially depend on the new CP-violating phases. We also
present the correlation between $\sin^2 \theta_{23}$ and the LIV parameter
$|a_{\alpha \beta}|$, as well as $\delta_{CP}$ and $|a_{\alpha \beta}|$.
