Isotropy-Violation Diagnostics for $B$-mode Polarization Foregrounds to
  the Cosmic Microwave Background
  Isotropy-violation statistics can highlight polarized galactic foregrounds
that contaminate primordial $B$-modes in the Cosmic Microwave Background (CMB).
We propose a particular isotropy-violation test and apply it to polarized
Planck 353 GHz data, constructing an map that indicates $B$-mode foreground
dust power over the sky. We build our main isotropy test in harmonic space via
the bipolar spherical harmonic basis, and our method helps us to identify the
least-contaminated directions. By this measure, there are regions of low
foreground in and around the BICEP field, near the South Galactic Pole, and in
the Northern Galactic Hemisphere. There is also a possible foreground feature
in the BICEP field. We compare our results to those based on the local power
spectrum, which is computed on discs using a version of the method of Planck
Int.~XXX (2016). The discs method is closely related to our isotropy-violation
diagnostic. We pay special care to the treatment of noise, including chance
correlations with the foregrounds. Currently we use our isotropy tool to assess
the cleanest portions of the sky, but in the future such methods will allow
isotropy-based null tests for foreground contamination in maps purported to
measure primordial $B$-modes, particularly in cases of limited frequency
coverage.
