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 are new.

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.