  In view of the recent measurement of the scintillation velocity for PSR
J0737-3039, we examine the complete set of constraints imposed on the pulsar B
natal kicks (magnitude and orientation) and predict the most favorable pulsar
kick velocity and spin tilt for both isotropic and polar kicks. Our analysis
takes into account both currently unknown parameters: the orientation of the
orbital plane on the sky (Omega) and the radial component of the systemic
velocity (V_r). Assuming that the system's peculiar velocity is entirely due to
the second supernova explosion, we find that the system may have crossed the
Galactic plane multiple times since the birth of the second neutron star and
that the post-supernova peculiar velocity could have been as high as 1200km/s.
We also confirm the absolute lower and upper limits on the physical parameters
derived in our earlier study. For specific combinations of the two unknown
parameters Omega and V_r, however, we find much tighter constraints on the
pre-supernova binary configuration and natal kicks imparted to pulsar B, as
well as on the age of system. Once Omega is measured in the coming year, it
will be straightforward to use the results presented here to further constrain
the natal kicks and the spin-tilt predictions. We complete our comprehensive
study and derive similar constraints and spin-tilt predictions for PSR
B1534+12, where the only free parameter is V_r. Lastly, for PSR B1913+16, we
update the progenitor and kick constraints using the measured pulsar spin tilt
and allowing for Roche-lobe overflow from the progenitor of the pulsar
companion.
