  We examine the impact of Lorentz Invariance Violation (LIV) in measuring the
octant of $\theta_{23}$ and CP phases in the context of the Deep Underground
Neutrino Experiment (DUNE). We consider the CPT-violating LIV parameters
involving $e - \mu$ ($a_{e\mu}$) and $e - \tau$ ($a_{e\tau}$) flavors, which
induce an additional interference term in neutrino and antineutrino appearance
probabilities. This new interference term depends on both the standard CP phase
$\delta$ and the new dynamical CP phase $\varphi_{e\mu}$/$\varphi_{e\tau}$,
giving rise to new degeneracies among ($\theta_{23}$, $\delta$, $\varphi$).
Taking one LIV parameter at-a-time and considering a small value of $|a_{e\mu}|
= |a_{e\tau}| = 5 \times 10^{-24}$ GeV, we find that the octant discovery
potential of DUNE gets substantially deteriorated for unfavorable combinations
of $\delta$ and $\varphi_{e\mu}$/$\varphi_{e\tau}$. The octant of $\theta_{23}$
can only be resolved at $3\sigma$ if the true value of $\sin^2\theta_{23}
\lesssim 0.42$ or $\gtrsim 0.62$ for any choices of $\delta$ and $\varphi$.
Interestingly, we also observe that when both the LIV parameters $a_{e\mu}$ and
$a_{e\tau}$ are present together, they cancel out the impact of each other to a
significant extent, allowing DUNE to largely regain its octant resolution
capability. We also reconstruct the CP phases $\delta$ and
$\varphi_{e\mu}$/$\varphi_{e\tau}$. The typical $1\sigma$ uncertainty on
$\delta$ is $10^{\circ}$ to $15^{\circ}$ and the same on
$\varphi_{e\mu}$/$\varphi_{e\tau}$ is $25^{\circ}$ to $30^{\circ}$ depending on
the choices of their true values.
