While commutativity of watermarking and encryption is a desirable feature in many application scenarios, it is hard to find robust watermarking schemes and secure ciphers which are able to commute with each other, because there are no visual features to use for embedding the mark in the encrypted domain. In the present paper we investigate if orthogonal maps, which form a large subclass of norm-preserving maps, are suitable for image encryption within the framework of a commutative watermarking-encryption (CWE) scheme. Specifically, we show that these maps, if used properly, have a much larger key space and leave a smaller statistical residue in the ciphertext than other norm-preserving maps like sign-bit encryption and permutation ciphers currently being used in CWE.