We investigate systematically whether accreting black hole systems are likely to reach global alignment of the black hole spin and its accretion disc with the binary plane. In low-mass X-ray binaries (LMXBs) there is only a modest tendency to reach such global alignment, and it is difficult to achieve fully: except for special initial conditions we expect misalignment of the spin and orbital planes by ~1 radian for most of the LMXB lifetime. The same is expected in high-mass X-ray binaries (HMXBs). A fairly close approach to global alignment is likely in most stellar-mass ultraluminous X-ray binary systems (ULXs) where the companion star fills its Roche lobe and transfers on a thermal timescale to a black hole of lower mass. These systems are unlikely to show orbital eclipses, as their emission cones are close to the hole's spin axis. This offers a potential observational test, as models for ULXs invoking intermediate-mass black holes do predict eclipses for ensembles of > ~10 systems. Recent observational work shows that eclipses are either absent or extremely rare in ULXs, supporting the picture that most ULXs are stellar-mass binaries with companion stars more massive than the accretor.

Accepted for publication in MNRAS in original form. 4 pages, 1 figure