  The kernel of the $\mathbb{Z}_2^{n}$ Dirac-Dunkl operator is examined. The
symmetry algebra $\mathcal{A}_{n}$ of the associated Dirac-Dunkl equation on
$\mathbb{S}^{n-1}$ is determined and is seen to correspond to a higher rank
generalization of the Bannai-Ito algebra. A basis for the polynomial
null-solutions of the Dirac-Dunkl operator is constructed. The basis elements
are joint eigenfunctions of a maximal commutative subalgebra of
$\mathcal{A}_{n}$ and are given explicitly in terms of Jacobi polynomials. The
symmetry algebra is shown to act irreducibly on this basis via raising/lowering
operators. A scalar realization of $\mathcal{A}_{n}$ is proposed.
