  In this paper, we prove that any nontrivial $\mathrm{SO}(k )\times
\mathrm{SO}(n+1-k)$-symmetric ancient compact noncollapsed solution of the mean
curvature flow agrees up to scaling and rigid motion with the
$\mathrm{O}(k)\times \mathrm{O}(n+1-k)$-symmetric ancient ovals constructed by
Hershkovits and the second author. This confirms a conjecture by
Angenent-Daskalopoulos-Sesum. On the other hand, for every $k\geq 2$ we also
construct a $(k-1)$-parameter family of uniformly $(k+1)$-convex ancient ovals
that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n+1-k)$-symmetric. This
gives counterexamples to a conjecture of Daskalopoulos.
