We consider the problem of nonlinear steady convection in a horizontal mushy layer rotating about a vertical axis. We analyse the stationary modes of convection in the form of two-dimensional oblique rolls and three-dimensional distorted patterns. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the two- and three-dimensional solutions to the weakly nonlinear problem by using a perturbation technique, and the stability of these solutions is investigated with respect to arbitrary three-dimensional disturbances. The results of the analyses in a particular range of values of the amplitude of convection indicate in particular that, over most of the range of values of the parameters, subcritical convection in the form of down-hexagons with down-flow at the cell centres and up-flow at the cell boundaries can be preferred over up-hexagonal convection, where the convective flow is upward at the cell centres and downward at the cell boundaries. For zero or very small values of ${\cal T}$ (${\cal T}\,{\ll}\,1$), which is the square root of a Taylor number, rolls are preferred over supercritical rectangles, while supercritical rectangles, which are characterized by an angle $\gamma$ of about $60^\circ$, are stable and preferred over the rolls for T above some value. Here, $\gamma$ or $180^\circ-\gamma$ are the angles between any two adjacent wavenumber vectors of a rectangular cell. For increasing values of T, these rectangles become subcritically unstable and are replaced by new supercritical rectangles of higher $\gamma$ values, and $\gamma$ increases with T until supercritical squares ($\gamma\,{=}\,90^\circ$) become stable. The significance and realizability of down-hexagons, rectangles and squares are found to be due to the interactions between the local solid fraction and the flow associated with the Coriolis term in the momentum–Darcy equation that are fully taken into account in the present study.