define Stokes parameters along the line of sight ${\hat{\bmath{n}}}$ using the basis vectors of a spherical polar coordinate system. We take the local $x$ axis to be $\hat{\bmath{\theta}}$ and the $y$ axis $-\hat{\bmath{\phi}}$; these form a right-handed triad with the radiation propagation direction $-{\hat{\bmath{n}}}$ which defines the local $z$ axis. The complex polarization $P \equiv Q + i U$ is then spin -2 [@newman66] and can properly be expanded in terms of spin-2 spherical harmonics (see e.g. @lewis02 for a review): $$\begin{aligned}
(Q\pm i U)({\hat{\bmath{n}}}) = \sum_{lm} (E_{lm} \mp i B_{lm}) {}_{\mp 2} Y_{lm}({\hat{\bmath{n}}}).
\label{eq:1}\end{aligned}$$ Reality of the Stokes parameters ensures that $E_{lm}^\ast = (-1)^m E_{l-m}$ and similarly for $B_{lm}$, while under parity transformations $E_{lm} \rightarrow (-1)^l E_{lm}$ (electric parity) but $B_{lm} \rightarrow (-1)^{l+1} B_{lm}$ (magnetic parity). The polarization power spectra are defined in statistically-isotropic and parity-respecting models by $$\langle E_{lm} E^*_{l'm'} \rangle = C_l^E \delta_{ll'} \delta_{mm'},
\quad \langle B_{lm} B^*_{l'm'} \rangle = C_l^B \delta_{ll'} \delta_{mm'}.
\label{eq:2}$$ The $E$-$B$ power vanishes (and $B$ is uncorrelated with the temperature anisotropies) by parity.

In pseudo-$C_l$ methods, one adopts a heuristic weighting of the complex polarization with a scalar function $w({\hat{\bmath{n}}})$. The weight is necessarily zero in those regions that are not observed, 