Two entwined problems have remained unresolved since pulsars were discovered nearly 50 years ago: the orientation of their polarized emission relative to the emitting magnetic field and the direction of putative supernova ``kicks relative to their rotation axes. The rotational orientation of most pulsars can be inferred only from the (``fiducial) polarization angle of their radiation, when their beam points directly at the Earth and the emitting polar fluxtube field is $parallel$ to the rotation axis. Earlier studies have been unrevealing owing to the admixture of different types of radiation (core and conal, two polarization modes), producing both $parallel$ or $perp$ alignments. In this paper we analyze the some 50 pulsars having three characteristics: core radiation beams, reliable absolute polarimetry, and accurate proper motions. The ``fiducial polarization angle of the core emission, we then find, is usually oriented $perp$ to the proper-motion direction on the sky. As the primary core emission is polarized $perp$ to the projected magnetic field in Vela and other pulsars where X-ray imaging reveals the orientation, this shows that the proper motions usually lie $parallel$ to the rotation axes on the sky. Two key physical consequences then follow: first, to the extent that supernova ``kicks are responsible for pulsar proper motions, they are mostly $parallel$ to the rotation axis; and second that most pulsar radiation is heavily processed by the magnetospheric plasma such that the lowest altitude ``parent core emission is polarized $perp$ to the emitting field, propagating as the extraordinary (X) mode.
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