  We consider the sine-Gordon and affine Toda field theories on the half-line
with classically integrable boundary conditions, and show that in the quantum
theory a remnant survives of the bulk quantized affine algebra symmetry
generated by non-local charges. The paper also develops a general framework for
obtaining solutions of the reflection equation by solving an intertwining
property for representations of certain coideal subalgebras of quantized affine
algebras.
