  The radiative decays of $b$-baryons facilitate the direct measurement of
photon helicity in $b\to s\gamma$ transitions thus serving as an important test
of physics beyond the Standard Model. In this paper we analyze the complete
angular distribution of ground state $b$-baryon ($\Lambda_{b}^{0}$ and
$\Xi_{b}^{-}$) radiative decays to multibody final states assuming an initially
polarized $b$-baryon sample. Our sensitivity study suggests that the photon
polarization asymmetry can be extracted to a good accuracy along with a
simultaneous measurement of the initial $b$-baryon polarization. With higher
yields of $b$-baryons, achievable in subsequent runs of the Large Hadron
Collider (LHC), we find that the photon polarization measurement can play a
pivotal role in constraining different new physics scenarios.
