We discuss a particularly symmetric model of neutrino mixings where, with good accuracy, the atmospheric mixing angle θ is maximal, θ=0 and the solar angle satisfies sinθ=1/3 (Harrison-Perkins-Scott (HPS) matrix). The discrete symmetry A is a suitable symmetry group for the realization of this type of model. We construct a model where the HPS matrix is exactly obtained in a first approximation without imposing ad hoc relations among parameters. The crucial issue of the required VEV alignment in the scalar sector is discussed and we present a natural solution of this problem based on a formulation with extra dimensions. We study the corrections from higher dimensionality operators allowed by the symmetries of the model and discuss the conditions on the cut-off scales and the VEVs in order for these corrections to be completely under control. Finally, the observed hierarchy of charged lepton masses is obtained by assuming a larger flavour symmetry. We also show that, under general conditions, a maximal θ can never arise from an exact flavour symmetry.