Summary: Let \(m,n\) be positive integers such that \(m<n\) and let \(G(n,m)\) be the Grassmann manifold of all \(m\)-dimensional subspaces of \(\mathbb{R}^n\). For \(V\in G(n,m)\) let \(\pi_V\) denote the orthogonal projection from \(\mathbb{R}^n\) onto \(V\). The following characterization of purely unrectifiable sets holds. Let \(A\) be an \({\mathcal H}^m\)-measurable subset of \(\mathbb{R}^n\) with \({\mathcal H}^m(A)<\infty\). Then \(A\) is purely \(m\)-unrectifiable if and only if there exists a null subset \(Z\) of the universal bundle \(\{(V,v)|V\in G(n,m),v\in V\}\) such that, for all \(P\in A\), one has \({\mathcal H}^{m(n-m)}(\{V\in G(n,m)|(V,\pi_V(P))\in Z\})>0\). One can replace “for all \(P\in A\)” by “for \({\mathcal H}^m\)-a.e. \(P\in A\)”.