  The algebraic structure of the rank two Racah algebra is studied in detail.
We provide an automorphism group of this algebra, which is isomorphic to the
permutation group of five elements. This group can be geometrically interpreted
as the symmetry of a folded icosidodecahedron. It allows us to study a class of
equivalent irreducible representations of this Racah algebra. They can be
chosen symmetric so that their transition matrices are orthogonal. We show that
their entries can be expressed in terms of Racah polynomials. This construction
gives an alternative proof of the recurrence, difference and orthogonal
relations satisfied by the Tratnik polynomials, as well as their expressions as
a product of two monovariate Racah polynomials. Our construction provides a
generalization of these bivariate polynomials together with their properties.
