What is the basis of gauge theory?

I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign signified. In other words, the underlying phenomenon remained constant, only the viewpoints changed, and therefore could be equated. But it seems that gauge theory is the opposite – identical observable quantities are seen, although the configurations of the underlying fields change. Is this right, or am I completely off-track?