Commutants of separately radial Toeplitz operators in several variables

If φ is a bounded separately radial function on the unit ball, the Toeplitz operator Tφ is diagonalizable with respect to the standard orthogonal basis of monomials on the Bergman space. Given such a φ, we characterize bounded functions ψ for which Tψ commutes with Tφ. Several examples are given to illustrate our results.