A cyclically permutable code is a binary block code of length \(n\) such that each codeword has \(n\) distinct cyclic shifts and such that no codeword can be obtained by one or more cyclic shifts of another codeword. Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple access (CDMA) radio systems and optical CDMA. In this paper the authors address the question of when a particular error-correcting code is equivalent (by permutation of the symbols) to a cyclically permutable code. The problem is introduced for simplex codes and a motivating example is given. In the final section the construction technique is illustrated initially for simplex codes (for which the complete answer is given) and is generalized to other cyclic codes.