The double-polarization observable E and helicity-dependent cross sections σ_{1/2}, σ_{3/2} have been measured for the photoproduction of π^{0} pairs off quasifree protons and neutrons at the Mainz MAMI accelerator with the Crystal Ball/TAPS setup. A circularly polarized photon beam was produced by bremsstrahlung from longitudinally polarized electrons and impinged on a longitudinally polarized deuterated butanol target. The reaction products were detected with an almost 4π covering calorimeter. The results reveal for the first time the helicity- and isospin-dependent structure of the γN→Nπ^{0}π^{0} reaction. They are compared to predictions from reaction models in view of nucleon resonance contributions and also to a refit of one model that predicted results for the proton and for the neutron target. The comparison of the prediction and the refit demonstrates the large impact of the new data.