The newly introduced signature of benzenoids (a sequence of six real numbers si with i = 6-1) shows the composition of the pi-electron partition by indicating the number of times all rings of the benzenoid are assigned 6. 5, 4, 3, 2, or 1 pi-electrons. It allows the introduction of a new ordering criterion for such polycyclic aromatic hydrocarbons by summing some of the terms in the signature. There is an almost perfect linear correlation between SUMS s(6) + s(5) and s(4) + s(3) for isomeric cata- or peri-fused benzenoids, so that one can sort such isomers according to ascending s(6) + s(5) or to descending s(4) + s(3) sums (the resulting ordering does not differ much and agrees with that based on increasing numbers of Clar sextets and of Kekule structures). Branched cata-condensed benzenoids have higher s(6) + s(5) sums than isomeric nonbranched systems. For nonisomeric peri-condensed benzenoids. both sums increase with increasing numbers of benzenoid rings and decrease with the number of internal carbon atoms. Other partial sums that have been explored are s(6) + s(5) + s(3) and s(6) + s(2) + s(1), and the last one appears to be more generally applicable as a parameter for the complexity of benzenoids and for ordering isomeric benzenoids.