  We study a subclass of K\"ahler-Einstein Fano polygons and how they behave
under mutation. The polygons of interest are K\"ahler-Einstein Fano triangles
and symmetric Fano polygons. In particular, we find an explicit bound for the
number of these polygons in an arbitrary mutation-equivalence class.
  An important mutation-invariant of a Fano polygon is its singularity content.
We extend the notion of singularity content and prove that it is still a
mutation-invariant. We use this to show that if two symmetric Fano polygons are
mutation-equivalent, then they are isomorphic. We further show that if two
K\"ahler-Einstein Fano triangles are mutation-equivalent, then they are
isomorphic. Finally, we show that if a symmetric Fano polygon is
mutation-equivalent to a K\"ahler-Einstein triangle, then they are isomorphic.
Thus, each mutation-equivalence class has at most one Fano polygon which is
either a K\"ahler-Einstein triangle or symmetric.
  A recent conjecture states that all K\"ahler-Einstein Fano polygons are
either triangles or are symmetric. We provide a counterexample $P$ to this
conjecture and discuss several of its properties. For instance, we compute
iterated barycentric transformations of $P$ and find that (a) the
K\"ahler-Einstein property is not preserved by the barycentric transformation,
and (b) $P$ is of strict type $B_2$. Finally, we find examples of
K\"ahler-Einstein Fano polygons which are not minimal.
