  We revisit the classification of polarization observables of the cosmic
microwave background. There exists a unified approach to the $3 \times 3$
density matrix by which intensity, linear and circular polarization are treated
on an equal footing. The representations of the rotation group on the sphere
contain certain right-left setup symmetries which have not been taken into
account before. Left-right symmetries revise the construction of invariants and
certain predictions based on symmetries. There are two true rotational
invariants mode by mode in polarization data. Independent of models of
cosmological perturbations, we emphasize methods to construct invariant
distributions in order to test predictions of isotropy, axial or parity
symmetries, and generate covariant and invariant statistics order by order.
