The first measurement is reported of the double-polarization observable G in the photoproduction of neutral pions off protons, covering the photon energy range from 620 to 1120 MeV and the full solid angle. G describes the correlation between the photon polarization plane and the scattering plane for protons polarized along the direction of the incoming photon. The observable is highly sensitive to contributions from baryon resonances. The new results are compared to the predictions from SAID, MAID, and BnGa partial wave analyses. In spite of the long-lasting efforts to understand γp→pπ(0) as the simplest photoproduction reaction, surprisingly large differences between the new data and the latest predictions are observed which are traced to different contributions of the N(1535) resonance with spin parity J(P)=1/2(-) and N(1520) with J(P)=3/2(-). In the third resonance region, where N(1680) with J(P)=5/2(+) production dominates, the new data are reasonably close to the predictions.