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We study the interplay between mass transfer, accretion and gravitationaltorques onto a black hole binary migrating in a self-gravitating, retrogradecircumbinary disc. A direct comparison with an identical prograde disc showsthat: (i) because of the absence of resonances, the cavity size is a factora(1+e) smaller for retrograde discs; (ii) nonetheless the shrinkage of acircular binary semi--major axis, a, is identical in both cases; (iii) acircular binary in a retrograde disc remains circular while eccentric binariesgrow more eccentric. For non-circular binaries, we measure the orbital decayrates and the eccentricity growth rates to be exponential as long as the binaryorbits in the plane of its disc. Additionally, for these co-planar systems, wefind that interaction (~ non--zero torque) stems only from the cavity edge plusa(1+e) in the disc, i.e. for dynamical purposes, the disc can be treated as aannulus of small radial extent. We find that simple 'dust' models in which thebinary- disc interaction is purely gravitational can account for all mainnumerical results, both for prograde and retrograde discs. Furthermore, wediscuss the possibility of an instability occurring for highly eccentricbinaries causing it to leave the disc plane, secularly tilt and converge to aprograde system. Our results suggest that there are two stable configurationsfor binaries in self-gravitating discs: the special circular retrograde caseand an eccentric (e~ 0.6) prograde configuration as a stable attractor.

Migration of massive black hole binaries in self--gravitating accretion discs: Retrograde versus prograde