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The following questions are germane to our understanding of gauge-(in)variantquantities and physical possibility: how are gauge transformations andspacetime diffeomorphisms understood as symmetries, in which ways are theysimilar, and in which are they different? To what extent are we justified inendorsing different attitudes -- nowadays called sophistication, haecceitism,and eliminativism -- towards each? This is the first of four papers taking upthis question.  This first paper will discuss notions of symmetries and isomorphisms thatwill be used in the remaining papers in the series. There are several suchnotions in the literature and the question of how they mesh with empiricaldiscernibility is a delicate one; even the orthodox view that symmetries areempirically unobservable (even in principle) has recently been challenged byBelot. Focusing on local field theories, I will provide a precise definition ofdynamical symmetries in terms of the space of states of the theory at hand. Iwill then apply the definition to Yang-Mills theories and general relativityand show that these symmetries correspond to automorphisms of `natural'geometric structures: the small diffeomorphisms of the spacetime manifold andthe small fiber-preserving diffeomorphisms of a fibered space. Finally, I willshow that these automorphisms can be given a passive gloss, since theycorrespond 1-1 to the coordinate transformations of the underlying manifolds.

Same-diff? Conceptual similarities between gauge transformations and diffeomorphisms. Part I: Symmetries and isomorphisms