Classifying Polarization Observables of the Cosmic Microwave Background

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.