Recently, Pitts (2014) has argued that the claim that first-class constraints generate gauge transformations (hereafter “orthodoxy”) fails even in electromagnetism, which is standardly taken to illustrate its correctness. Independently, Barbour and Foster (2008) have argued that a key presupposition of the primary argument for orthodoxy (due to Dirac, 1964) is not satisfied in the important case of reparameterization-invariant theories. In assessing these claims, one needs to distinguish between, (i) transformations that relate points of phase space that represent the same instantaneous state and (ii) transformations that map curves on phase space to curves that represent the same history. Pitts shows that arbitrary first-class constraints fail to generate transformations of type (ii), but leaves untouched the orthodox position concerning (i). Barbour and Foster show that we have no reason to regard transformations generated by Hamiltonian constraints as type (i), but that leaves open that they may be regarded as transformations of type (ii). I will discuss whether the latter possibility allows one to reconcile applying Dirac’s constrained quantization procedure to the Hamiltonian constraint and interpreting the Wheeler–DeWitt wavefunction as representing genuine change.