Turbulent wakes past bluff bodies commonly present asymmetric flow states reminiscent of bifurcations in the laminar regime. Understanding the sensitivity of these states to flow forcing is crucial to the modelling and control of flow symmetry properties. In this study, the near wake of a rectangular bluff body in proximity to a wall is disturbed by the use of passive devices located between the model and the wall, upstream of the massive flow separation occurring at the blunt trailing edges. Due to the proximity to the boundary, the wake initially presents wall-normal asymmetry and a negative wall-normal pressure gradient along the base. The application of disturbances with variable size, however, sets flow symmetry along the wall-normal plane, leading to the intermittent spanwise wake reversals reported recently in the literature. A further increase in the size of perturbation suppresses wake switching, and wall-normal asymmetry is recovered, but with a positive wall-normal pressure gradient. The dynamical features of this bifurcation scenario can be retrieved using two coupled symmetry-breaking models for spanwise and wall-normal pressure gradients. This confirms the high sensitivity of the separated flow to external perturbations. More importantly, the results unify observations of the bluff-body wake topologies covered in previous investigations.