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 0737-3039A, 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.