The twisted q-Yangians are coideal subalgebras of the quantum affine algebra
associated with gl(N). We prove a classification theorem for finite-dimensional
irreducible representations of the twisted q-Yangians associated with the
symplectic Lie algebras sp(2n). The representations are parameterized by their
highest weights or by their Drinfeld polynomials. In the simplest case of sp(2)
we give an explicit description of all the representations as tensor products
of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem
for the quantum affine algebra and for the twisted q-Yangians. We also
reproduce a proof of the classification theorem for finite-dimensional
irreducible representations of the quantum affine algebra by relying on its
R-matrix presentation.