The problem of understanding the origin of the observed pattern of neutrino mixing and, more generally, the origin of flavour is among the biggest problems in particle physics. Driven by the measured values of the neutrino mixing parameters, we adopt symmetry approach to neutrino mixing, based on the assumption of existence of a non-Abelian discrete flavour symmetry. The most distinct feature of this approach is correlations between the neutrino mixing angles and CP-violating phases, which are referred to as neutrino mixing sum rules. We first consider all types of the residual symmetries of the charged lepton and neutrino mass matrices for which such correlations are expected and derive the corresponding sum rules for the cosine of the Dirac phase $\delta$. Using these sum rules, we obtain predictions for $\cos\delta$ in the cases of several discrete flavour symmetries. Next, we concentrate on a scenario in which the main contribution to neutrino mixing arises from the neutrino sector in the form of highly symmetric mixing patterns. We explore possible charged lepton corrections to these patterns required to reconstitute their compatibility with experimental data. In the context of the proposed DUNE and T2HK facilities, we study (i) the compatibility of these symmetric mixing patterns with present neutrino oscillation data, and (ii) the potential of these experiments to discriminate between various symmetric patterns.