  In this paper, we consider the large time asymptotic behavior of solutions to
systems of two cubic nonlinear Klein-Gordon equations in one space dimension.
We classify the systems by studying the quotient set of a suitable subset of
systems by the equivalence relation naturally induced by the linear
transformation of the unknowns. It is revealed that the equivalence relation is
well described by an identification with a matrix. In particular, we
characterize some known systems in terms of the matrix and specify all systems
equivalent to them. An explicit reduction procedure from a given system in the
suitable subset to a model system, i.e., to a representative, is also
established. The classification also draws our attention to some model systems
which admit solutions with a new kind of asymptotic behavior. Especially, we
find new systems which admit a solution of which decay rate is worse than that
of a solution to the linear Klein-Gordon equation by logarithmic order.
