  We consider an unified model, called ancestor model, associated with twisted
trigonometric $R$ matrix which model leads to several descendant integrable
lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary
operators compatible with integrability are introduced to this model.
Reflection and dual reflection equations to ensure integrability of the system
are shown to be same as the untwisted case. It follows that underlying symmetry
of the ancestor model with integrable boundaries is identified with the
q-deformed analogue of Onsager's symmetry. The transfer matrix and its related
mutually commuting quantities are expressed in terms of an abelian subalgebra
in the $q$-Onsager algebra. It is illustrated that the generalized McCoy-Wu
model with general open boundaries enjoys this symmetry.
