It is known that automorphic subsets are generalizations of cwatsets. In this paper we show that an automorphic subset is the translation of some cwatset, and therefore that each automorphic subset is internally isomorphic to a cwatset.
Lepinski, Matthew, "Automorphic Subsets of the n-dimensional Cube are Translations of Cwatsets" (1999). Mathematical Sciences Technical Reports (MSTR). 56.