The onset of convection in a rotating cylindrical annulus with sloping conical boundaries is studied in the case where this slope increases with the radius. The critical modes assume the form of drifting spiralling columns attached to the inner cylindrical wall at moderate and large Prandtl numbers, but they become attached to the outer wall at low Prandtl numbers. These latter ‘equatorially attached’ modes are multicellular at intermediate rotation rates. Through a perturbation analysis which is validated by a numerical code, we show that all equatorially attached modes are quasi-inertial modes and analyse the physical mechanisms leading to multicells. This is done for both stress-free and no-slip boundary conditions. At finite amplitudes the convection generates a Reynolds stress which leads to the development of a mean zonal flow, and a geometrical analysis of the mechanisms leading to this zonal flow is presented. The influence of Ekman friction on the zonal flow is also studied.