The natural orbital functional theory admits two unique representations in the orbital space. On the one hand, we have the natural orbitals themselves that minimize the energy functional, and which afford for a diagonal one-particle reduced density matrix but not for a diagonal Lagrangian orbital energy multipliers matrix. On the other hand, since it is possible to reverse the situation but only once the energy minimization has been achieved, we have the so-called canonical representation, where the Lagrangian orbital energy multipliers matrix is diagonal but the one-particle reduced density matrix is not. Here it is shown that the former representation, the natural orbital representation, accounts nicely for the quadrupole bond character of the ground states of C2, BN, CB(-), and CN(+), and for the double bond order character of the isovalent (1)Σg (+) state of Si2. Similarly, the canonical orbital representation accounts correctly for the ionization spectra of all these species.