  We study the effect of pulsar rotation on timing of binary pulsars, with
particular emphasis on the double pulsar system J0737-3039. Special
relativistic aberration due to the orbital motion of pulsar changes both the
longitude and colatitude of the emission direction with respect to the pulsar
spin axis. The former gives rise to a shift of the arrival time of the pulse
centroid (this is the conventional ``longitudinal'' aberration delay), the
latter results in a distortion (contraction or dilation) of the pulse profile
on the orbital timescale. The amplitude of pulse distortion depends inversely
on the variation of polarization position angle across the pulse. For small
angle between the pulsar magnetic and spin axes, as inferred for PSR
J0737-3039A from polarimetric observations, the pulse distortion is significant
(\sim 1%) and the associated ``latitudinal'' aberration delay is much larger
than the longitudinal one. By monitoring the arrival time of separate pulse
components as a function of pulsar orbital phase, the latitudinal aberration
delay may be easily measured with the current timing precision, thus
constraining the spin geometry of the system. The latitudinal delay can also be
detected by monitoring system's orbital parameters on the geodetic precession
timescale. Because of the near edge-on orbital orientation of the PSR
J0737-3039 system, general relativistic bending of pulsar A's radio beam near
its superior conjunction also introduces spin-dependent time delays of similar
order of magnitude as the aberration delays. In addition, light bending splits
the pulse profile into two variable components, corresponding to two
gravitationally lensed images of the source. Detection of lensing effects is
challenging, but may be possible with existing technology.
