A count of Kekulé structures for all 1812 distinct fullerene isomers of C₆₀ shows that 20 isomers surpass the count of 12500 for icosahedral C₆₀, and demonstrates the lack of correlation between molecular-orbital indices of stability and raw Kekulé counts for fullerenes. Analysis of Kekulé structures in terms of benzenoid, cyclopentenoid and cyclopentadienoid rings reveals the source of the stability of icosahedral C₆₀ in a localised model to be the fact that uniquely amongst the 1812 structural isomers it has a Fries Kekulé structure where all hexagons contain 3 double bonds and all pentagons none.