The central properties of the usual variational formulations of EINSTEIN's general theory of relativity are sketched and a more general variational formulation is introduced, which is more appropriate to the geometric foundations of the theory. In particular, this formulation leads to the BIANCHI identities in their non-contracted form following in consequence of a general invariance property. This invariance property of the general variational principle relates to transformations (not, in general, coordinate transformations) that contain arbitrary fifth (or third) order tensor “generators”. These results can be interpreted as implying that the variational principle introduced here admits an “extended principle of general covariance” (i.e., a covariance principle more general than the usual principle relating to general coordinate covariance). Some of the formal implications of these results as well as their connection with general coordinate covariance are discussed briefly. In particular these results point to the existence of a fundamental transformation theory connecting all of the RIEMANNian spacetimes of general relativity.