Interpreting map-based $E$/$B$ spectral properties of CMB foregrounds
Map-space $E$/$B$ decompositions of linear polarization are attractive for foreground and CMB analyses because they separate parity families: $B$-family patterns directly contaminate primordial tensor searches, while $E$-family patterns trace coherent Galactic structures. However, the $E$/$B$ transform is not fully local and can induce apparent spectral complexity even when the underlying sky is spectrally simple in $\underline{P}=Q+iU$. We quantify this effect for synchrotron emission using complex log--Taylor and moment expansions for $\underline{P}$, its spin-preserving projections $\underline{P}_E$ and $\underline{P}_B$, and its more standard scalar projections $E$ and $B$. We relate the coefficients of these expansions to physical mechanisms such as line-of-sight mixing, synchrotron ageing, and Faraday effects. Using simple sky models, we show how $\underline{P}_E$ and $\underline{P}_B$ reorganize the spectral behaviour of $\underline{P}$ into parity families with a clear geometric meaning. They retain interpretable amplitudes and angles and satisfy the closure relation $\underline{P}=\underline{P}_E+\underline{P}_B$, which extends to all moment orders. By contrast, scalar quantities such as $|E|$ and $|B|$ show larger induced variability of effective spectral parameters and enhanced spectral complexity, while $E+iB$ lacks interpretable polarization amplitude and angle. Finally, we present simple CMB-oriented applications: three-frequency diagnostics to test whether the sky is better described by a power law in $P$ or by separate effective laws in $(P_E,P_B)$, and idealized ILC and masking examples showing that the preferred map-space field depends on where foreground simplicity and residual contamination reside. This framework provides practical diagnostics for choosing foreground modelling, cleaning, and masking strategies in Galactic and CMB $B$-mode analyses.