  We study origin, consequences and testability of a hypothesis of `partial
$\mu$-$\tau$' reflection symmetry. This symmetry predicts $ |U_{\mu
i}|=|U_{\tau i}|~(i=1,2,3) $ for a single column of the leptonic mixing matrix
$U$. Depending on whether this symmetry holds for the first or second column of
$U$ different correlations between $\theta_{23}$ and $ \delta_{CP} $ can be
obtained. This symmetry can be obtained using discrete flavour symmetries. In
particular, all the subgroups of SU(3) with 3-dimensional irreducible
representation which are classified as class C or D can lead to partial
$\mu$-$\tau$ reflection symmetry. We show how the predictions of this symmetry
compare with the allowed area in the $\sin^2\theta_{23} - \delta_{CP}$ plane as
obtained from the global analysis of neutrino oscillation data. Furthermore, we
study the possibility of testing these symmetries at the proposed DUNE and
Hyper-Kamiokande (HK) experiments (T2HK, T2HKK), by incorporating the
correlations between $\theta_{23}$ and $ \delta_{CP}$ predicted by the
symmetries. We find that when simulated data of DUNE and HK is fitted with the
symmetry predictions, the $\theta_{23}-\delta_{CP}$ parameter space gets
largely restricted near the CP conserving values of $ \delta_{CP} $. Finally,
we illustrate the capability of these experiments to distinguish between the
two cases leading to partial $\mu-\tau$ symmetry namely $|U_{\mu1}| = |U_{\tau
1}|$ and $|U_{\mu 2}| = |U_{\tau 2}|$.
