  We analyze the effect of symmetrization in the theory of multiple orthogonal
polynomials. For a symmetric sequence of type II multiple orthogonal
polynomials satisfying a high-term recurrence relation, we fully characterize
the Weyl function associated to the corresponding block Jacobi matrix as well
as the Stieltjes matrix function. Next, from an arbitrary sequence of type II
multiple orthogonal polynomials with respect to a set of d linear functionals,
we obtain a total of d+1 sequences of type II multiple orthogonal polynomials,
which can be used to construct a new sequence of symmetric type II multiple
orthogonal polynomials. Finally, we prove a Favard-type result for certain
sequences of matrix multiple orthogonal polynomials satisfying a matrix
four-term recurrence relation with matrix coefficients.
