We study the correlation energy, the effective anisotropy parameter, and quantum fluctuations of the pseudospin magnetization in bilayer quantum Hall systems at total filling factor ν=1 by means of exact diagonalizations of the Hamiltonian in the spherical geometry. We compare exact-diagonalization results for the ground-state energy with finite-size Hartree-Fock values. In the ordered ground-state phase at small layer separations the Hartree-Fock data compare reasonably with the exact results. Above the critical layer separation, however, the Hartree-Fock findings still predict an increase in the ground-state energy, while the exact ground-state energy is in this regime independent of the layer separation indicating the decoupling of layers and the loss of spontaneous phase coherence between them. We also find accurate values for the pseudospin anisotropy constant, whose dependence of the layer separation provides another very clear indication for the strong interlayer correlations in the ordered phase and shows an inflection point at the phase boundary. Finally, we discuss the possibility of interlayer correlations in biased systems even above the phase boundary for the balanced case. Certain features of our data for the pseudospin anisotropy constant as well as for quantum fluctuations of the pseudospin magnetization are not inconsistent with the occurrence of this effect. However, it appears to be rather weak at least in the limit of vanishing tunneling amplitude.