In this paper we study a stratum of integrable cubic vector fields with a saddle singularity having symmetry, i.e. symmetric with respect to two axes. Our perspective is from the point of view of invariant algebraic curves of the systems. We study the global geometry of such systems. We give the bifurcation diagram of the phase portraits of the vector fields. All bifurcations correspond to bifurcations of invariant algebraic curves. We next try to link this bifurcation diagram with the one given by Rousseau and Schlomiuk for integrable cubic vector fields with a centre singularity having symmetry.