  We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are
Lyapunov stable both for backward and forward iterations. We prove they must
admit a dominated splitting and show that under some hypothesis they must be
the whole manifold. As a consequence of our results we also prove that in
dimension 2 the class must be the whole manifold and in dimension 3, these
classes must have nonempty interior. Many results on Lyapunov stable homoclinic
classes for $C^1$-generic diffeomorphisms are also deduced.
