  Generalized Baxter's TQ-relations and the QQ-system are systems of algebraic
relations in the category O of representations of the Borel subalgebra of the
quantum affine algebra U_q(g^), which we established in our earlier works
arXiv:1308.3444 and arXiv:1606.05301. In the present paper, we conjecture a
family of analogous relations labeled by elements of the Weyl group W of g, so
that the original relations correspond to the identity element. These relations
are closely connected to the W-symmetry of q-characters established in
arXiv:2211.09779. We prove these relations for all w in W if g has rank two,
and we prove the extended TQ-relations if w is a simple reflection. We also
generalize our results and conjectures to the shifted quantum affine algebras.
