Over a field of characteristic zero, a family of polynomials $(p_n)$ is computed by matrix-symmetric formulas of polynomial orbit size if, and only if, there is a constant $t$ such that every $p_n$ is a linear combination of homomorphism polynomials $\hom_{F,n}$ of patterns $F$ of treedepth at most $t$. This is a result of Dwivedi, Pago, and Seppelt (STOC 2026). Symmetric formulas have rigidifications of bounded support depth, so it follows from the characterisation of bounded-depth matrix-symmetric circuits, since bounded $\Delta$-forestwidth for bounded $\Delta$ amounts to bounded treedepth.