  In this paper a twofold inverse problem for orthogonal matrix functions in
the Wiener class is considered. The scalar-valued version of this problem was
solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem
is reduced to an invertibility condition on an operator that is defined using
the Hankel and Toeplitz operators associated to the Wiener class functions that
comprise the data set of the inverse problem. It is also shown that in this
case the solution is unique. Special attention is given to the case that the
Hankel operator of the solution is a strict contraction and the case where the
functions are matrix polynomials.
