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Unpaired image-to-image translation has attracted significant interest due to the invention of CycleGAN, a method which utilizes a combination of adversarial and cycle consistency losses to avoid the need for paired data. It is known that the CycleGAN problem might admit multiple solutions, and our goal in this paper is to analyze the space of exact solutions and to give perturbation bounds for approximate solutions. We show theoretically that the exact solution space is invariant with respect to automorphisms of the underlying probability spaces, and, furthermore, that the group of automorphisms acts freely and transitively on the space of exact solutions. We examine the case of zero `pure' CycleGAN loss first in its generality, and, subsequently, expand our analysis to approximate solutions for `extended' CycleGAN loss where identity loss term is included. In order to demonstrate that these results are applicable, we show that under mild conditions nontrivial smooth automorphisms exist. Furthermore, we provide empirical evidence that neural networks can learn these automorphisms with unexpected and unwanted results. We conclude that finding optimal solutions to the CycleGAN loss does not necessarily lead to the envisioned result in image-to-image translation tasks and that underlying hidden symmetries can render the result utterly useless.