  In this article we consider a quaternionic inner form $G$ of a $p$-adic
classical group defined over a non-archimedian local field of odd residue
characteristic. We construct all full self-dual semisimple characters for $G$
and we classify their intertwining classes using endo-parameters. Further we
prove an intertwining and conjugacy theorem for self-dual semisimple
characters. We give the formulas for the set of intertwiners between self-dual
semisimple characters. We count all $G$-intertwining classes of self-dual
semisimple characters which lift to the same $\tilde{G}$-intertwining class of
a semisimple character for the ambient general linear group $\tilde{G}$ for
$G$.
