The Nπ0π0 decays of positive-parity N* and Δ* resonances at about 2 GeV are studied at ELSA by photoproduction of two neutral pions off protons. The data reveal clear evidence for several intermediate resonances: Δ(1232), N(1520)3/2-, and N(1680)5/2+, with spin parities JP=3/2+, 3/2-, and 5/2+. The partial wave analysis (within the Bonn-Gatchina approach) identifies N(1440)1/2+ and the N(ππ)S wave (abbreviated as Nσ here) as further isobars and assigns the final states to the formation of nucleon and Δ resonances and to nonresonant contributions. We observe the known Δ(1232)π decays of Δ(1910)1/2+, Δ(1920)3/2+, Δ(1905)5/2+, Δ(1950)7/2+, and of the corresponding spin-parity series in the nucleon sector, N(1880)1/2+, N(1900)3/2+, N(2000)5/2+, and N(1990)7/2+. For the nucleon resonances, these decay modes are reported here for the first time. Further new decay modes proceed via N(1440)1/2+π, N(1520)3/2-π, N(1680)5/2+π, and Nσ. The latter decay modes are observed in the decay of N* resonances and at most weakly in Δ* decays. It is argued that these decay modes provide evidence for a 3-quark nature of N* resonances rather than a quark-diquark structure.