  Understanding and interpretation of the dynamics of ultrarelativistic plasma
is a challenge, which calls for the development of methods for \textit{in situ}
probing the plasma dynamical characteristics. We put forward a new method,
harnessing polarization properties of $\gamma$-photons emitted from a
non-pre-polarized plasma irradiated by a circularly polarized pulse. We show
that the angular pattern of $\gamma$-photon linear polarization is explicitly
correlated with the dynamics of the radiating electrons, which provides
information on the laser-plasma interaction regime. Furthermore, with the
$\gamma$-photon circular polarization originating from the electron radiative
spin-flips, the plasma susceptibility to quantum electrodynamical processes is
gauged. Our study demonstrates that the polarization signal of emitted
$\gamma$-photons can be a versatile information source, which would be
beneficial for the research fields of laser-driven plasma, accelerator science,
and laboratory astrophysics.
