Some conceptual issues concerning general invariant theories, with special emphasis on general relativity, are analyzed. The common assertion that observables must be required to be gauge invariant is examined in the light of the role played by a system of observers. Some features of the reduction of the gauge group are discussed, including the fact that in the process of a partial gauge fixing the reduction at the level of the gauge group and the reduction at the level of the variational principle do not commute. Distinctions between the mathematical and the physical concept of gauge symmetry are discussed and illustrated with examples. The limit from general relativity to special relativity is considered as an example of a gauge group reduction that is allowed in some specific physical circumstances. Whether and when the Poincar\'e group must be considered as a residual gauge group will come out as a result of our analysis, that applies, in particular, to asymptotically flat spaces.
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