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We study origin, consequences and testability of a hypothesis of `partial$\mu$-$\tau$' reflection symmetry. This symmetry predicts $ |U_{\mui}|=|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 beobtained. This symmetry can be obtained using discrete flavour symmetries. Inparticular, all the subgroups of SU(3) with 3-dimensional irreduciblerepresentation which are classified as class C or D can lead to partial$\mu$-$\tau$ reflection symmetry. We show how the predictions of this symmetrycompare with the allowed area in the $\sin^2\theta_{23} - \delta_{CP}$ plane asobtained from the global analysis of neutrino oscillation data. Furthermore, westudy the possibility of testing these symmetries at the proposed DUNE andHyper-Kamiokande (HK) experiments (T2HK, T2HKK), by incorporating thecorrelations between $\theta_{23}$ and $ \delta_{CP}$ predicted by thesymmetries. We find that when simulated data of DUNE and HK is fitted with thesymmetry predictions, the $\theta_{23}-\delta_{CP}$ parameter space getslargely restricted near the CP conserving values of $ \delta_{CP} $. Finally,we illustrate the capability of these experiments to distinguish between thetwo cases leading to partial $\mu-\tau$ symmetry namely $|U_{\mu1}| = |U_{\tau1}|$ and $|U_{\mu 2}| = |U_{\tau 2}|$.

Exploring Partial $\mu$-$\tau$ Reflection Symmetry at DUNE and Hyper-Kamiokande