A specific family of spanwise-localised invariant solutions of plane Couette flow exhibits homoclinic snaking, a process by which spatially localised invariant solutions of a nonlinear partial differential equation smoothly grow additional structure at their fronts while undergoing a sequence of saddle-node bifurcations. Homoclinic snaking is well understood in the context of simpler pattern-forming systems such as the one-dimensional Swift–Hohenberg equation with cubic-quintic nonlinearity. The Swift–Hohenberg solutions closely resemble the snaking solutions of plane Couette flow, yet this remarkable resemblance and the mechanisms supporting homoclinic snaking within the three-dimensional Navier–Stokes equations remain to be fully understood. Studies of Swift–Hohenberg revealed the central importance of discrete symmetries for homoclinic snaking to be supported by an equation. We therefore study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking in three-dimensional plane Couette flow for flow modifications that break symmetries of the flow. We demonstrate that wall-normal suction modifies the bifurcation structure of three-dimensional plane Couette solutions in the same way a symmetry-breaking quadratic term modifies solutions of the one-dimensional Swift–Hohenberg equation. These modifications are related to the breaking of the discrete rotational symmetry. At large amplitudes of the symmetry-breaking wall-normal suction the connected snakes-and-ladders structure is destroyed. Previously unknown solution branches are created and can be parametrically continued to vanishing suction. This yields new localised solutions of plane Couette flow that exist in a wide range of Reynolds numbers.