The implications of the principles of general and manifest covariance, together with those of the objectivity principle, are considered for the purpose of establishing a DeDonder-Weyl-type Hamiltonian variational formulation for classical general relativity. Based on the analysis of the Einstein-Hilbert variational principle, it is shown that only synchronous variational principles permit the construction of fully 4-tensor Lagrangian and Hamiltonian theories of this type. In addition, the possible validity of an extended Hamiltonian formulation in which Lagrangian variables include also the Ricci tensor is investigated and shown to occur provided the classical cosmological constant is non-vanishing.