© 2018 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. 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.