  Interactions between a supermassive black hole binary and the surrounding
accretion disc can both assist the binary inspiral and align the black hole
spins to the disc angular momentum. While binary migration is due to
angular-momentum transfer within the circumbinary disc, the spin-alignment
process is driven by the mass accreting on to each black hole. Mass transfer
between different disc components thus couples the inspiral and the alignment
process together. Mass is expected to leak through the cavity cleared by the
binary, and preferentially accretes on to the lighter (secondary) black hole
which orbits closer to the disc edge. Low accretion rate on to the heavier
(primary) black hole slows the alignment process down. We revisit the problem
and develop a semi-analytical model to describe the coupling between gas-driven
inspiral and spin alignment, finding that binaries with mass ratio q<~0.2
approach the gravitational-wave driven inspiral in differential misalignment:
light secondaries prevent primaries from aligning. Binary black holes with
misaligned primaries are ideal candidates for precession effects in the
strong-gravity regime and may suffer from moderately large (~1500 km/s) recoil
velocities.
