  We introduce the notion of a cylindrical bialgebra, which is a
quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a
universal solution of a generalized reflection equation, yielding an action of
cylindrical braid groups on tensor products of its representations. We prove
that new examples of such universal K-matrices arise from quantum symmetric
pairs of Kac-Moody type and depend upon the choice of a pair of generalized
Satake diagrams. In finite type, this yields a refinement of a result obtained
by Balagovi\'c and Kolb, producing a family of non-equivalent solutions
interpolating between the quasi-K-matrix originally due to Bao and Wang and the
full universal K-matrix. Finally, we prove that this construction yields formal
solutions of the generalized reflection equation with a spectral parameter in
the case of finite-dimensional representations over the quantum affine algebra
$U_qL\mathfrak{sl}_2$.
