  This paper examines and strengthens the Cuntz-Thomsen picture of equivariant
Kasparov theory for arbitrary second-countable locally compact groups, in which
elements are given by certain pairs of cocycle representations between
C*-dynamical systems. The main result is a stable uniqueness theorem that
generalizes a fundamental characterization of ordinary $KK$-theory by Lin and
Dadarlat-Eilers. Along the way, we prove an equivariant Cuntz-Thomsen picture
analog of the fact that the equivalence relation of homotopy agrees with the (a
priori stronger) equivalence relation of stable operator homotopy. The results
proved in this paper will be employed as the technical centerpiece in
forthcoming work of the authors to classify certain amenable group actions on
Kirchberg algebras by equivariant Kasparov theory.
