  We present the partition function of the refined Chern-Simons theory on $S^3$
with arbitrary A,B,C,D gauge algebra in terms of multiple sine functions. For B
and C cases this representation is novel. It allows us to conjecture duality to
some refined and orientifolded versions of the topological string on the
resolved conifold, and carry out the detailed identification of different
contributions. The free energies for D and C algebras possess the usual halved
contribution from the A theory, i.e. orientable surfaces, and contributions of
non-orientable surfaces with one cross-cup, with opposite signs, similar as for
the non-refined theories. However, in the refined case, both theories possess
in addition a non-zero contribution of orientable surfaces with two cross-cups.
In particular, we observe a trebling of the K\"ahler parameter, in the sense of
a refinement and world-sheet (i.e. the number of cross-cups) dependent quantum
shift. For B algebra the contribution of Klein bottles is zero, as is the case
in the non-refined theory, and the one-cross-cup terms differ from the D and C
cases. For the (refined) constant maps terms of these theories we suggest a
modular-invariant representation, which leads to natural topological string
interpretation. We also calculate some non-perturbative corrections.
