  Sometimes it is thought that the mere fact that two mathematical structures are intertransformable is all that is needed for the transformation to be a gauge transformation and for the differences between the two structures to correspond to nothing physical. Since the transformation is invertible, the essential fact is that any property of the first structure will have a correlated property in the second; and any property of the second will have a correlated property in the first. That means that the two structures are, informally speaking, perfect mathematical images of each other and each could stand in for the other in any formal application.