We introduce a class of systems holding parity-time (PT) symmetry locally, whereas being globally P symmetric. The potential, U = U(vertical bar r vertical bar), fulfills PT symmetry with respect to periodically distributed points r(0) : U(vertical bar r(0) + r vertical bar) = U*(vertical bar r(0) - r vertical bar) being r(0) not equal 0. We show that such systems hold unusual properties arising from the merging of the two different symmetries, leading to a strong field localization and enhancement at the double-symmetry center, r = 0, when the coupling of outward to inward propagating waves is favored. We explore such general potentials in one and two dimensions, which could have actual realizations combining gain-loss and index modulations in nanophotonic structures. In particular, we show how to render a broad aperture vertical-cavity surface-emitting laser into a bright and narrow beam source, as a direct application.