We explore a systematic approach to the analysis of primordial non-Gaussianity using fluctuations in the temperature and polarization of the cosmic microwave background (CMB). Following Munshi & Heavens, we define a set of power spectra as compressed forms of the bispectrum and trispectrum derived from CMB temperature and polarization maps; these spectra compress the information content of the corresponding full multispectra and can be useful in constraining early Universe theories. We generalize the standard pseudo-Cl estimators in such a way that they apply to these spectra involving both spin-0 and spin-2 fields, developing explicit expressions, which can be used in the practical implementation of these estimators. While these estimators are suboptimal, they are nevertheless unbiased and robust and hence can provide useful diagnostic tests at a relatively small computational cost. We next consider approximate inverse-covariance weighting of the data and construct a set of near-optimal estimators based on that approach. Instead of combining all available information from the entire set of mixed bi- or tri-spectra, that is, multispectra describing both temperature and polarization information, we provide analytical constructions for individual estimators, associated with particular multispectra. The bias and scatter of these estimators can be computed using Monte Carlo techniques. Finally, we provide estimators, which are completely optimal for arbitrary scan strategies and involve inverse-covariance weighting; we present the results of an error analysis performed using a Fisher-matrix formalism at both the one-point and two-point levels.