  We study some global aspects of the bifurcation of an equivariant family of
volume-contracting vector fields on the three-dimensional sphere. When part of
the symmetry is broken, the vector fields exhibit Bykov cycles. Close to the
symmetry, we investigate the mechanism of the emergence of heteroclinic
tangencies coexisting with transverse connections. We find persistent suspended
horseshoes accompanied by attracting periodic trajectories with long periods.
