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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, whose solutions remarkably well resemble the snaking solutions of plane Couette flow. We study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking for flow modifications that break symmetries of plane Couette 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 number.