Be stars are rapidly spinning B stars surrounded by an outflowing disc of gas in Keplerian rotation. Be star/X-ray binary systems contain a Be star and a neutron star. They are found to have non-zero eccentricities and there is evidence that some systems have a misalignment between the spin axis of the star and the spin axis of the binary orbit. The eccentricities in these systems are caused by a kick to the neutron star during the supernova that formed it. Such kicks would also give rise to misalignments. In this paper, we investigate the extent to which the same kick distribution can give rise to both the observed eccentricity distribution and the observed misalignments. We find that a Maxwellian distribution of velocity kicks with a low velocity dispersion, σk≈ 15 km s−1, is consistent with the observed eccentricity distribution but is hard to reconcile with the observed misalignments, typically i≥ 25°. Alternatively, a higher velocity kick distribution, σk= 265 km s−1, is consistent with the observed misalignments but not with the observed eccentricities, unless post-supernova circularization of the binary orbits has taken place. We discuss briefly how this might be achieved.