  We showed in the first paper of this series that the generic $C_2^1$-cofactor
matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we
obtain a combinatorial characterization of independence in this matroid. This
solves the cofactor counterpart of the combinatorial characterization problem
for the rigidity of generic 3-dimensional bar-joint frameworks. We use our
characterization to verify that the counterparts of conjectures of Dress (on
the rank function) and Lov\'{a}sz and Yemini (which suggested a sufficient
connectivity condition for rigidity) hold for this matroid.
