There are many possible VB diagrams for C60. Many of them are still symmetric. The bead model I showed in the previous post corresponds to the kekule structure of single hexagon rotated by one beads, which leads to a fusion of three pentagons into a trefoil. One can also perform this operation on the remaining 9 pentagons at suitable 3-fold axes to produce an interesting pattern which looks like a tiling of four trefoils on a sphere.
I made the following two bead models of C60 to show this pattern. (left column: north hemisphere; right column: south hemisphere)
It is easier to see this pattern with four trefoil tiles using a schematic plot.
(I would be happy to recommend this chiral pattern with T symmetry on a football for the FIFA next time!)