$symVP$ consists of the families spanned by homomorphism polynomials of patterns of bounded treewidth. For large $n$, the support theorem bounds the support size of a rigidified circuit, and the bounded-depth characterisation (with the support depth as depth bound) applies; for the finitely many remaining $n$, matrix-invariance and the general homomorphism spanning theorem suffice. Conversely, a common depth bound for the finitely many patterns in each linear combination gives circuits via the bounded-depth characterisation, while finitely many exceptional indices are covered by symmetric ABPs.