A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using graph-theoretical and finite-state-machine constructions. These are the mother graph and the Hoey-Sloane machine. In this paper, we leverage the inherent symmetry of the above constructions for the purpose of finding new permutiples from old. Such results also help us to see previous work through a new lens.