The two pages linked below gives the results of optimally solving
all symmetric positions of the cube, first in the half-turn metric
and then in the quarter-turn metric. The half-turn metric results
confirm Radu and Kociemba's results; the quarter-turn metric results
For insight into symmetry as it applies to the cube, I can't write
anything better than Herbert
Kociemba's page on symmetry and the cube.
Results in the half-turn metric.
Results in the quarter-turn metric.