Theoretical and numerical insight is gained into the epsilon-I relations between the Kohn-Sham orbital energies epsilon(i) and relaxed vertical ionization potentials (VIPs) I-j, which provide an analog of Koopmans' theorem for density functional theory. The Kohn-Sham orbital energy epsilon(i) has as leading term -n(i)I(i)-Sigma(jis an element ofOmegas)(i)n(j)I(j), where I-i is the primary VIP for ionization (phi(i))(-1) with spectroscopic factor (proportional to the intensity in the photoelectron spectrum) n(i) close to 1, and the set Omega(s)(i) contains the VIPs I-j that are satellites to the (phi(i))(-1) ionization, with small but non-negligible n(j). In addition to this "average spectroscopic structure" of the epsilon(i) there is an electron-shell step structure in epsilon(i) from the contribution of the response potential v(resp). Accurate KS calculations for prototype second- and third-row closed-shell molecules yield valence orbital energies -epsilon(i), which correspond closely to the experimental VIPs, with an average deviation of 0.08 eV. The theoretical relations are numerically investigated in calculations of the components of the epsilon-I relations for the H-2 molecule, and for the molecules CO, HF, H2O, HCN. The derivation of the epsilon-I relations employs the Dyson orbitals (the n(i) are their norms). A connection is made between the KS and Dyson orbital theories, allowing the spin-unrestricted KS xc potential to be expressed with a statistical average of individual xc potentials for the Dyson spin-orbitals as leading term. Additional terms are the correction v(c,kin,sigma) due to the correlation kinetic effect, and the "response" v(resp,sigma), related to the correction to the energy of (N-1) electrons due to the correlation with the reference electron. (C) 2003 American Institute of Physics.