  The quasifree $\overrightarrow{\gamma} d\to\pi^0n(p)$ photon beam asymmetry,
$\Sigma$, has been measured at photon energies, $E_\gamma$, from 390 to 610
MeV, corresponding to center of mass energy from 1.271 to 1.424 GeV, for the
first time. The data were collected in the A2 hall of the MAMI electron beam
facility with the Crystal Ball and TAPS calorimeters covering pion
center-of-mass angles from 49 to 148$^\circ$. In this kinematic region,
polarization observables are sensitive to contributions from the $\Delta
(1232)$ and $N(1440)$ resonances. The extracted values of $\Sigma$ have been
compared to predictions based on partial-wave analyses (PWAs) of the existing
pion photoproduction database. Our comparison includes the SAID, MAID, and
Bonn-Gatchina analyses; while a revised SAID fit, including the new $\Sigma$
measurements, has also been performed. In addition, isospin symmetry is
examined as a way to predict $\pi^0n$ photoproduction observables, based on
fits to published data in the channels $\pi^0p$, $\pi^+n$, and $\pi^-p$.
