The relativistic double neutron star binary PSR J0737-3039 shows clear evidence of orbital phase-dependent wind-companion interaction, both in radio and X-rays. In this paper we present the results of timing analysis of PSR J0737-3039 performed during 2006 and 2011 XMM-Newton Large Programs that collected ~20,000 X-ray counts from the system. We detected pulsations from PSR J0737-3039A (PSR A) through the most accurate timing measurement obtained by XMM-Newton so far, the spin period error being of 2x10^-13 s. PSR As pulse profile in X-rays is very stable despite significant relativistic spin precession that occurred within the time span of observations. This yields a constraint on the misalignment between the spin axis and the orbital momentum axis Delta_A ~6.6^{+1.3}_{-5.4} deg, consistent with estimates based on radio data. We confirmed pulsed emission from PSR J0737-3039B (PSR B) in X-rays even after its disappearance in radio. The unusual phenomenology of PSR Bs X-ray emission includes orbital pulsed flux and profile variations as well as a loss of pulsar phase coherence on time scales of years. We hypothesize that this is due to the interaction of PSR As wind with PSR Bs magnetosphere and orbital-dependent penetration of the wind plasma onto PSR B closed field lines. Finally, the analysis of the full XMM-Newton dataset provided evidences of orbital flux variability (~7%) for the first time, involving a bow-shock scenario between PSR As wind and PSR Bs magnetosphere.
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