  Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced
by Terng to find higher flows of the generalized sine-Gordon equation. Twisted
$\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important
classes of twisted hierarchies. In this paper, interesting first and higher
flows of twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are explicitly
derived, the associated submanifold geometry is investigated and a unified
treatment of the inverse scattering theory is provided.
