  We show that the monodromy of the trigonometric Casimir connection on the
tensor product of evaluation modules of the Yangian Ysl_2 is described by the
quantum Weyl group operators of the quantum loop algebra U_h(Lsl_2). The proof
is patterned on the second author's computation of the monodromy of the
rational Casimir connection for sl_n via the dual pair (gl_k,gl_n), and rests
ultimately on the Etingof-Geer-Schiffmann computation of the monodromy of the
trigonometric KZ connection. It relies on two new ingredients: an affine
extension of the duality between the R-matrix of U_h(sl_k) and the quantum Weyl
group element of U_h(sl_2), and a formula expressing the quantum Weyl group
action of the coroot lattice of SL_2 in terms of the commuting generators of
U_h(Lsl_2). Using this formula, we define quantum Weyl group operators for the
quantum loop algebra U_h(Lgl_2) and show that they describe the monodromy of
the trigonometric Casimir connection on a tensor product of evaluation modules
of the Yangian Ygl_2
