[en] We evaluate the polarization observables I^S and I^C for the reaction γp→π⁰ηp, using a chiral unitary framework developed earlier. The I^S and I^C observables have been recently measured for the first time by the CBELSA/TAPS Collaboration. The theoretical predictions of I^S and I^C, given for altogether 18 angle-dependent functions, are in good agreement with the measurements. Also, the asymmetry dΣ/d cos θ evaluated here agrees with the data. We show the importance of the Δ(1700)D₃₃ -resonance and its S -wave decay into ηΔ(1232). The result can be considered as a further confirmation of the dynamical nature of this resonance. At the highest energies, deviations of the predictions from the data start to become noticeable, which leaves room for additional processes and resonances such as a Δ(1940)D₃₃. We also point out how to further improve the calculation. (orig.)