  The first measurement is reported of the double-polarization observable G in
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 {\gamma}p -> p{\pi} 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) 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.
