  The double-polarization observable $E$ and helicity-dependent cross sections
$\sigma_{1/2}$, $\sigma_{3/2}$ have been measured for the photoproduction of
$\pi^0$ pairs off quasi-free 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 $4\pi$ covering calorimeter. The results reveal
for the first time the helicity- and isospin-dependent structure of the $\gamma
N\rightarrow N\pi^0\pi^0$ 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 demonstrate the large impact of
the new data.
