We derive the global bifurcation diagram of a three-parameter family of cubic Liénard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is. of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.