Our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in S³. The cycle involves two saddle-foci of different type and is structurally stable within the class of (Z₂⊕Z₂)-symmetric vector fields. The cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations. Gradually breaking the symmetry in a two-parameter family we get a wide range of dynamical behaviour: an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; n-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov type. We also show that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension 2 and takes place arbitrarily close to the symmetric cycle.