  We study the space of biinvariants and zonal spherical functions associated
to quantum symmetric pairs in the maximally split case. Under the obvious
restriction map, the space of biinvariants is proved isomorphic to the Weyl
group invariants of the character group ring associated to the restricted
roots. As a consequence, there is either a unique set, or an (almost) unique
two-parameter set of Weyl group invariant quantum zonal spherical functions
associated to an irreducible symmetric pair. Included is a complete and
explicit list of the generators and relations for the left coideal subalgebras
of the quantized enveloping algebra used to form quantum symmetric pairs.
