  We explore a systematic approach to the analysis of primordial
non-Gaussianity using fluctuations in temperature and polarization of the
Cosmic Microwave Background (CMB). Following Munshi & Heavens (2009), 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-C_l
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 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 trispectra, i.e
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 level.
