  We consider the following question: How much of the combinatorial structure
determining properties of $\overline{\mathcal{M}_{0, n}}$ is ``intrinsic'' and
how much new information do we obtain from using properties specific to this
space? Our approach is to study the effect of the $S_n$-action. Apart from
being a natural action to consider, it is known that this action does not
extend to other wonderful compactifications associated to the $A_{n - 2}$
hyperplane arrangement. We find the differences in intersection patterns of
faces on associahedra and permutohedra which characterize the failure to extend
to other compactifications and show that this is reflected by most terms of
degree $\ge 2$ of the cohomology/Chow ring.
  Even from a combinatorial perspective, terms of degree 1 are more naturally
related to geometric properties. In particular, imposing $S_n$-invariance
implies that many of the log concave sequences obtained from degree 1
Hodge--Riemann relations (and all of them for $n \le 2000$) on the Chow ring of
$\overline{\mathcal{M}_{0, n}}$ can be restricted to those with a special
recursive structure. A conjectural result implies that this is true for all
$n$. Elements of these sequences can be expressed as polynomials in quantum
Littlewood--Richardson coefficients multiplied by terms such as partition
components, factorials, and multinomial coefficients. After dividing by
binomial coefficients, polynomials with these numbers as coefficients can be
interepreted in terms of volumes or resultants. Finally, we find a connection
between the geometry of $\overline{\mathcal{M}_{0, n}}$ and higher degree
Hodge--Riemann relations of other rings via Toeplitz matrices.
