Neutrino- and antineutrino-oxygen neutral-current quasielasticlike interactions are measured at Super-Kamiokande using nuclear deexcitation γ rays to identify signal-like interactions in data from a 14.94(16.35)×1020 protons-on-target exposure of the T2K neutrino (antineutrino) beam. The measured flux-averaged cross sections on oxygen nuclei are ⟨σν−NCQE⟩=1.70±0.17(stat.)−0.38+0.51(syst.)×10−38 cm2/oxygen with a flux-averaged energy of 0.82 GeV and ⟨σν¯−NCQE⟩=0.98±0.16(stat.)−0.19+0.26(syst.)×10−38 cm2/oxygen with a flux-averaged energy of 0.68 GeV, for neutrinos and antineutrinos, respectively. These results are the most precise to date, and the antineutrino result is the first cross section measurement of this channel. They are compared with various theoretical predictions. The impact on evaluation of backgrounds to searches for supernova relic neutrinos at present and future water Cherenkov detectors is also discussed.