  We report the first detection of galactic spiral structure by means of
thermal emission from magnetically aligned dust grains. Our 89 $\mu$m
polarimetric imaging of NGC 1068 with the High-resolution Airborne Wideband
Camera/Polarimeter (HAWC+) on NASA's Stratospheric Observatory for Infrared
Astronomy (SOFIA) also sheds light on magnetic field structure in the vicinity
of the galaxy's inner-bar and active galactic nucleus (AGN). We find
correlations between the 89 $\mu$m magnetic field vectors and other tracers of
spiral arms, and a symmetric polarization pattern as a function of the
azimuthal angle arising from the projection and inclination of the disk field
component in the plane of the sky. The observations can be fit with a
logarithmic spiral model with pitch angle of $16.9^{+2.7}_{-2.8}$$^{\circ}$ and
a disk inclination of $48\pm2^{\circ}$. We infer that the bulk of the
interstellar medium from which the polarized dust emission originates is
threaded by a magnetic field that closely follows the spiral arms. Inside the
central starburst disk ($<1.6$ kpc), the degree of polarization is found to be
lower than for far-infrared sources in the Milky Way, and has minima at the
locations of most intense star formation near the outer ends of the inner-bar.
Inside the starburst ring, the field direction deviates from the model,
becoming more radial along the leading edges of the inner-bar. The polarized
flux and dust temperature peak $\sim 3-6$" NE of the AGN at the location of a
bow shock between the AGN outflow and the surrounding interstellar medium, but
the AGN itself is weakly polarized ($< 1$%) at both 53 and 89 \um.
