Summary: We examine an alternative formulation of the gauge principle in which the emphasis is shifted from the symmetry transformations to their generators. We show that the gauge principle can be entirely reformulated in terms of promoting constants of motion-which generate rigid symmetries-to constraints-which generate gauge symmetries. In our exposition we first explain the basic philosophy on mechanical systems, and then with the help of De Donder-Weyl formalism we extend our scenario also to a field-theoretical setting. To illustrate this, we demonstrate our method in numerous examples, including the massive relativistic particle, the Nambu-Goto closed string and relativistic field theory.