We study coupled Dirac operators on the Euclidean space Rd, which are associated with unitary connections which extend to the sphere S^d (in a possibly non-trivial bundle). We define Fredholm extensions of these operators to Sobolev type completions, and we prove that the kernel and the cokernel of these extensions can be identified with the corresponding spaces of harmonic spinors on the sphere. For q, p > 1 related by 1/d + 1/q = 1/p result provides bounded L^p → L^q operators which invert the coupled Dirac operators modulo operators of finite rank.