  We investigate the consequences of $\mu-\tau$ reflection symmetry in presence
of a light sterile neutrino for the $3+1$ neutrino mixing scheme. We discuss
the implications of total $\mu-\tau$ reflection symmetry as well partial
$\mu-\tau$ reflection symmetry. For the total $\mu-\tau$ reflection symmetry we
find values of $\theta_{23}$ and $\delta$ remains confined near $\pi/4$ and
$\pm \pi/2$ respectively. The current allowed region for $\theta_{23}$ and
$\delta$ in case of inverted hierarchy lies outside the area preferred by the
total $\mu-\tau$ reflection symmetry. However, interesting predictions on the
neutrino mixing angles and Dirac CP violating phases are obtained considering
partial $\mu-\tau$ reflection symmetry. We obtain predictive correlations
between the neutrino mixing angle $\theta_{23}$ and Dirac CP phase $\delta$ and
study the testability of these correlations at the future long baseline
experiment DUNE. We find that while the imposition of $\mu-\tau$ reflection
symmetry in the first column admit both normal and inverted neutrino mass
hierarchy, demanding $\mu-\tau$ reflection symmetry for the second column
excludes the inverted hierarchy. Interestingly, the sterile mixing angle
$\theta_{34}$ gets tightly constrained considering the $\mu-\tau$ reflection
symmetry in the fourth column. We also study consequences of $\mu-\tau$
reflection symmetry for the Majorana phases and neutrinoless double beta decay.
