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The lack of power of large--angle CMB anisotropies is known to increase itsstatistical significance at higher Galactic latitudes, where a string--inspiredpre--inflationary scale $\Delta$ can also be detected. Considering the Planck2015 data, and relying largely on a Bayesian approach, we show that the effectis mostly driven by the \emph{even}--$\ell$ harmonic multipoles with $\ell\lesssim 20$, which appear sizably suppressed in a way that is robust withrespect to Galactic masking, along with the corresponding detections of$\Delta$. On the other hand, the first \emph{odd}--$\ell$ multipoles are onlysuppressed at high Galactic latitudes. We investigate this behavior indifferent sky masks, constraining $\Delta$ through even and odd multipoles, andwe elaborate on possible implications. We include low--$\ell$ polarization datawhich, despite being noise--limited, help in attaining confidence levels ofabout 3 $\sigma$ in the detection of $\Delta$. We also show by direct forecaststhat a future all--sky $E$--mode cosmic--variance--limited polarization surveymay push the constraining power for $\Delta$ beyond 5 $\sigma$.

The Evens and Odds of CMB Anomalies