  Reflection equation algebras and related U_q(g)-comodule algebras appear in
various constructions of quantum homogeneous spaces and can be obtained via
transmutation or equivalently via twisting by a cocycle. In this paper we
investigate algebraic and representation theoretic properties of such so called
`covariantized' algebras, in particular concerning their centres, invariants,
and characters. Generalising M. Noumi's construction of quantum symmetric pairs
we define a coideal subalgebra B_f of U_q(g) for each character f of a
covariantized algebra. The locally finite part F_l(U_q(g)) of U_q(g) with
respect to the left adjoint action is a special example of a covariantized
algebra. We show that for each character f of F_l(U_q(g)) the centre Z(B_f)
canonically contains the representation ring Rep(g) of the semisimple Lie
algebra g. We show moreover that for g=sl_n(C) such characters can be
constructed from any invertible solution of the reflection equation and hence
we obtain many new explicit realisations of Rep(sl_n(C)) inside U_q(sl_n(C)).
As an example we discuss the solutions of the reflection equation corresponding
to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in C^{2m}.
