  We establish a symmetrization procedure in a context of general orthogonal
expansions associated with a second order differential operator $L$, a
`Laplacian'. Combined with a unified conjugacy scheme furnished in our earlier
article it allows, via a suitable embedding, to associate a
differential-difference `Laplacian' $\mathbb{L}$ with the initially given
orthogonal system of eigenfunctions of $L$, so that the resulting extended
conjugacy scheme has the natural classical shape. This means, in particular,
that the related `partial derivatives' decomposing $\mathbb{L}$ are
skew-symmetric in an appropriate $L^2$ space and they commute with Riesz
transforms and conjugate Poisson integrals. The results shed also some new
light on the question of defining higher order Riesz transforms for general
orthogonal expansions.
