  In this work, we give some criteria that allow us to decide when two
sequences of matrix-valued orthogonal polynomials are related via a Darboux
transformation and to build explicitly such transformation. In particular, they
allow us to see when and how any given sequence of polynomials is Darboux
related to a diagonal matrix of classic orthogonal polynomials. We also explore
the notion of Darboux-irreducibility and study some sequences that are not a
Darboux transformation of classical orthogonal polynomials.
