  Matrix-valued spherical functions related to the quantum symmetric pair for
the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and
studied in detail. The quantum symmetric pair is given in terms of a quantised
universal enveloping algebra with a coideal subalgebra. The matrix-valued
spherical functions give rise to matrix-valued orthogonal polynomials, which
are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For
these matrix-valued orthogonal polynomials a number of properties are derived
using this quantum group interpretation: the orthogonality relations from the
Schur orthogonality relations, the three-term recurrence relation and the
structure of the weight matrix in terms of Chebyshev polynomials from tensor
product decompositions, the matrix-valued Askey-Wilson type $q$-difference
operators from the action of the Casimir elements. A more analytic study of the
weight gives an explicit LDU-decomposition in terms of continuous
$q$-ultraspherical polynomials. The LDU-decomposition gives the possibility to
find explicit expressions of the matrix entries of the matrix-valued orthogonal
polynomials in terms of continuous $q$-ultraspherical polynomials and $q$-Racah
polynomials.
