  This paper builds on the previous paper arXiv:math/0612730 by the author,
where a relationship between Zhedanov's algebra AW(3) and the double affine
Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was
established. It is shown here that the spherical subalgebra of this DAHA is
isomorphic to AW(3) with an additional relation that the Casimir operator
equals an explicit constant. A similar result with q-shifted parameters holds
for the antispherical subalgebra. Some theorems on centralizers and centers for
the algebras under consideration will finally be proved as corollaries of the
characterization of the spherical and antispherical subalgebra.
