  Using high-resolution data from the Galactic Arecibo L-Band Feed Array HI
(GALFA-HI) survey, we show that linear structure in Galactic neutral hydrogen
(HI) correlates with the magnetic field orientation implied by Planck 353 GHz
polarized dust emission. The structure of the neutral interstellar medium is
more tightly coupled to the magnetic field than previously known. At high
Galactic latitudes, where the Planck data are noise-dominated, the HI data
provide an independent constraint on the Galactic magnetic field orientation,
and hence the local dust polarization angle. We detect strong
cross-correlations between template maps constructed from estimates of dust
intensity combined with either HI-derived angles, starlight polarization
angles, or Planck 353 GHz angles. The HI data thus provide a new tool in the
search for inflationary gravitational wave B-mode polarization in the cosmic
microwave background, which is currently limited by dust foreground
contamination.
