  The double pulsar system PSR J0737-3039A/B is a highly relativistic double
neutron star (DNS) binary, with a 2.4-hour orbital period. The low mass of the
second-formed NS, as well the low system eccentricity and proper motion, point
to a different evolutionary scenario compared to other known DNS systems. We
describe analysis of the pulse profile shape over 6 years of observations, and
present the resulting constraints on the system geometry. We find the recycled
pulsar in this system, PSR J0737-3039A, to have a low misalignment between its
spin and orbital angular momentum axes, with a 68.3% upper limit of 6.1
degrees, assuming emission from both magnetic poles. This tight constraint
lends credence to the idea that the supernova that formed the second pulsar was
relatively symmetric, possibly involving electron-capture onto an O-Ne-Mg core.
