The n-qubit Pauli group and its normalizer, the n-qubit Clifford group, have applications in quantum error correction and device characterization. Recent applications have used the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.