  We describe bivariate polynomial sequences orthogonal to a symmetric weight
function in terms of several bivariate polynomial sequences orthogonal with
respect to Christoffel transformations of the initial weight under a quadratic
transformation. We analyze the construction of a symmetric bivariate orthogonal
polynomial sequence from a given one, orthogonal to a weight function defined
on the positive plane. In this description plays an important role a sort of
Backlund type matrix transformations for the involved three term matrix
coefficients. We take as a case study relations between symmetric orthogonal
polynomials defined on the ball and on the simplex.
