  We prove that the vanishings of the modified diagonal cycles of Gross and
Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals
$\text{pt}^{\{1,\ldots,n\}\setminus A}\times\Delta_A$ in the rational Chow ring
of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\ldots,n\}$. Our
results generalize to arbitrary symmetric classes in place of the diagonal in
$X^m$, and with different types of inclusions
$A^\bullet(X^m)_{\mathbb{Q}}^{S_m} \hookrightarrow
A^\bullet(X^n)_{\mathbb{Q}}$.
  The combinatorial heart of this paper, which may be of independent interest,
is showing the $\mathbb{Z}$-linear relations between elementary symmetric
polynomials $e_k(x_{a_1},\ldots,x_{a_m}) \in \mathbb{Z}[x_1,\ldots,x_n]$ are
generated by the $S_n$-translates of a certain alternating sum over the facets
of a hyperoctahedron.
