  Using the theory introduced by Casper and Yakimov, we investigate the
structure of algebras of differential and difference operators acting on matrix
valued orthogonal polynomials (MVOPs) on $\mathbb{R}$, and we derive algebraic
and differential relations for these MVOPs. A particular case of importance is
that of MVOPs with respect to a matrix weight of the form $W(x)=e^{-v(x)}e^{xA}
e^{xA^\ast}$ on the real line, where $v$ is a scalar polynomial of even degree
with positive leading coefficient and $A$ is a constant matrix.
