  The helicity-dependent cross sections for the photoproduction of $\pi^0\eta$
pairs have been measured for the first time. The experiment was performed at
the tagged photon facility of the Mainz MAMI accelerator with the combined
Crystal Ball - TAPS calorimeter. The experiment used a polarized deuterated
butanol target and a circularly polarized photon beam. This arrangement allowed
the $\sigma_{1/2}$ (photon and target spin antiparallel) and $\sigma_{3/2}$
(parallel spins) components to be measured for quasi-free production of
$\pi^0\eta$ pairs off protons and neutrons. The main finding is that the two
helicity components contribute identically, within uncertainties, for both
participant protons and neutrons. The absolute couplings for protons and
neutrons are also identical. This means that nucleon resonances contributing to
this reaction in the investigated energy range have almost equal
electromagnetic helicity couplings, $A_{1/2}^{n,p}$ and $A_{3/2}^{n,p}$.
Identical couplings for protons and neutrons are typical for $\Delta$
resonances and identical $A_{1/2}$ and $A_{3/2}$ components are only possible
for $J\geq 3/2$ states, which constrains possible contributions of nucleon
resonances.
