A theory of a class of singular integrals on starlike Lipschitz surfaces in Rn is established. The class of singular integrals forms an operator algebra identical to the class of bounded holomorphic Fourier multipliers, as well as to the Cauchy-Dunford bounded holomorphic functional calculus of the spherical Dirac operator. The study proposes a new method inducing Clifford holomorphic functions from holomorphic functions of one complex variable, by means of which problems on the sphere are reduced to those on the unit circle. © 2001 Academic Press.