  Polarized Compton scattering on the deuteron is studied in nuclear effective
field theory. A set of tensor structures is introduced to define 12 independent
Compton amplitudes. The scalar and vector amplitudes are calculated up to
${\cal O}((Q/\Lambda)^2)$ in low-energy power counting. Significant
contribution to the vector amplitudes is found to come from the spin-orbit type
of relativistic corrections. A double-helicity dependent cross section
$\Delta_1 \sigma = (\sigma_{+1-1}-\sigma_{+1+1})/2$ is calculated to the same
order, and the effect of the nucleon isoscalar spin-dependent polarizabilities
is found to be smaller than the effect of isoscalar spin-independent ones.
Contributions of spin-independent polarizabilities are investigated in various
asymmetries, one of which has as large as 12 (26) percent effect at the
center-of-mass photon energy 30 (50) MeV.
