  We report here the first polarimetric measurements of the pulsars in the
J0737-3039 binary neutron star system using the Green Bank Telescope. We
conclude both that the primary star (A) has a wide hollow cone of emission,
which is an expected characteristic of the relatively open magnetosphere given
its short spin period, and that A has a small angle between its spin and
magnetic dipole axes, $4\pm 3$ degrees. This near alignment of axes suggests
that A's wind pressure on B's magnetosphere will depend on orbital phase. This
variable pressure is one mechanism for the variation of flux and profile shape
of B with respect to the orbital phase that has been reported. The response of
B to the A wind pressure will also depend on the particular side of its
magnetosphere facing the wind at the spin phase when B is visible. This is a
second possible mechanism for variability. We suggest that B may have its spin
axis aligned with the orbital angular momentum owing to A's wind torque that
contributes to its spindown. Monitoring the pulsars while geodetic precession
changes spin orientations will provide essential evidence to test detailed
theoretical models. We determine the Rotation Measures of the two stars to be
$-112.3\pm 1.5$ and $-118\pm 12$ rad m$^{-2}$.
