Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple-access (CDMA) radio systems and optical CDMA. The usual emphasis is on finding constant weight cyclically permutable codes with the maximum number of codewords. In this paper the question of when a particular error-correcting code is equivalent (by permutation of the symbols) to a cyclically permutable code is addressed. The problem is introduced for simplex codes and a motivating example is given. In the final section it is shown that the construction technique may be applied in general to cyclic codes.