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.