We systematically investigate the effects of high supernova kick velocities on the orbital parameters of post-supernova neutron-star binaries. Using Monte Carlo simulations, we determine the post-supernova distributions of orbital parameters (orbital period, eccentricity, system velocity, spin inclination, ratio of spin to orbital angular momentum) for progenitors of high-mass X-ray binaries (HMXBs) and low-mass X-ray binaries (LMXBs). With the recent distribution of pulsar birth velocities by Lyne and Lorimer, only about 27 per cent of massive systems remain bound after the supernova, of which similar to 26 per cent immediately experience dynamical mass transfer and possibly merge to become Thorne-Zytkow objects. The correlations between various orbital parameters can be compared with observational samples to yield information about supernova kick velocities and pre-supernova orbital-period distributions. After the supernova, the spins of most stars in massive systems have large inclinations with respect to their orbital axes, and a significant fraction of systems (similar to 20 per cent) contain stars with retrograde spins. This may have important implications for the interpretation of those HMXBs that seem to have tilted, 'precessing' accretion discs. We estimate that the spin angular momentum in the massive components of most HMXBs is a significant fraction (0.1-0.4) of the total orbital angular momentum. Therefore spin-orbit coupling effects may be important in many HMXBs. In the case of low-mass companions, we find that similar to 19 per cent of systems remain bound after the supernova, of which similar to 57 per cent experience immediate dynamical mass transfer. The systems that survive as binaries and become LMXB progenitors attain a very large system velocity of 180 +/- 80 km s(-1) after the supernova. There is a relatively tight correlation between the eccentricity and the post-supernova orbital period in these systems. All LMXBs with post-supernova periods longer than a few days initially have very large eccentricities. This may suggest that there should be a special class of LMXBs with periodic outbursts, of which Cir X-1 may be an extreme representative. We also use the results of these calculations to simulate the sky distributions of HMXBs and LMXBs. The simulated distributions agree with observed samples. Most importantly, the distribution of Galactic LMXBs is consistent with an ordinary Galactic disc population that has been widened because of large supernova kicks and does not require a special population of progenitors. The observed LMXB distribution does not provide a strong constraint on the age of LMXBs since the supernova, although there may be a weak hint that they are relatively young, with ages less than similar to 10(8) yr.
