  In this paper, we prove that any bubble-sheet oval for the mean curvature
flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the
$\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by
Hershkovits and the fourth author, or belongs to the one-parameter family of
$\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the
third and fourth author. In particular, this seems to be the first instance of
a classification result for geometric flows that are neither cohomogeneity-one
nor selfsimilar.
