  Global behavior of solutions is studied for the nonlinear Klein-Gordon
equation with a focusing power nonlinearity and a damping term in the energy
space on the Euclidean space. We give a complete classification of solutions
into 5 types of global behavior for all initial data in a small neighborhood of
each superposition of two ground states (2-solitons) with the opposite signs
and sufficient spatial distance. The neighborhood contains, for each sign of
the ground state, the manifold with codimension one in the energy space,
consisting of solutions that converge to the ground state at time infinity. The
two manifolds are joined at their boundary by the manifold with codimension two
of solutions that are asymptotic to 2-solitons moving away from each other. The
connected union of these three manifolds separates the rest of the neighborhood
into the open set of global decaying solutions and that of blow-up. The main
ingredient in the proof is a difference estimate on two solutions starting near
2-solitons and asymptotic to 1-solitons. The main difficulty is in controlling
the direction of the two unstable modes attached to 2-solitons, while the
soliton interactions are not uniformly integrable in time. It is resolved by
showing that the non-scalar part of the interaction between the unstable modes
is uniformly integrable due to the symmetry of the equation and the 2-solitons.
