  A higher rank generalization of the (rank one) Racah algebra is obtained as
the symmetry algebra of the Laplace-Dunkl operator associated to the
$\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra of
the generic superintegrable model on the $n$-sphere. Bases of Dunkl harmonics
are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases
consist of joint eigenfunctions of maximal Abelian subalgebras of the higher
rank Racah algebra. A method to obtain expressions for both the connection
coefficients between these bases and the action of the symmetries on these
bases is presented.
