The double pulsar system PSR J0737-3039A/B is a double neutron star binary, with a 2.4-hour orbital period, which has allowed measurement of relativistic orbital perturbations to high precision. The low mass of the second-formed neutron star, as well as the low system eccentricity and proper motion, point to a different evolutionary scenario compared to most other known double neutron star systems. We describe analysis of the pulse profile shape over 6 years of observations, and present the

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... lting constraints on the system geometry. We find the recycled pulsar in this system, PSR J0737-3039A, to be a near-orthogonal rotator, with an average separation between its spin and magnetic axes of 90 +/- 11 +/- 5 deg. Furthermore, we find a mean 95% upper limit on the misalignment between its spin and orbital angular momentum axes of 3.2 deg, assuming that the observed emission comes 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.