The Evens and Odds of CMB Anomalies
  The lack of power of large--angle CMB anisotropies is known to increase its
statistical significance at higher Galactic latitudes, where a string--inspired
pre--inflationary scale $\Delta$ can also be detected. Considering the Planck
2015 data, and relying largely on a Bayesian approach, we show that the effect
is mostly driven by the \emph{even}--$\ell$ harmonic multipoles with $\ell
\lesssim 20$, which appear sizably suppressed in a way that is robust with
respect to Galactic masking, along with the corresponding detections of
$\Delta$. On the other hand, the first \emph{odd}--$\ell$ multipoles are only
suppressed at high Galactic latitudes. We investigate this behavior in
different sky masks, constraining $\Delta$ through even and odd multipoles, and
we elaborate on possible implications. We include low--$\ell$ polarization data
which, despite being noise--limited, help in attaining confidence levels of
about 3 $\sigma$ in the detection of $\Delta$. We also show by direct forecasts
that a future all--sky $E$--mode cosmic--variance--limited polarization survey
may push the constraining power for $\Delta$ beyond 5 $\sigma$.
