We study the (1S) radiative decays to γπ+π- and γK+K- using data recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy e+e- collider at center-of-mass energies at the (2S) and (3S) resonances. The (1S) resonance is reconstructed from the decay (nS)→π+π-(1S), n=2, 3. Branching fraction measurements and spin-parity analyses of (1S) radiative decays are reported for the I=0 S-wave and f2(1270) resonances in the π+π- mass spectrum, the f2′(1525) and f0(1500) in the K+K- mass spectrum, and the f0(1710) in both.