Summary: We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which \(E(k {\rightarrow} 0) {\sim} {k}^{2}\), where \(E\) is the energy spectrum and \(k\) the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor \({\Phi}_{ij} (\mathbf{k})\) at small \(k\). These are \({M}_{\parallel}= {\Phi}_{\parallel}({k}_{\parallel}= 0, {k}_{\perp}{\rightarrow} 0)\) and \({M}_{\perp}= 2{\Phi}_{\perp}({k}_{\parallel}= 0, {k}_{\perp}{\rightarrow} 0)\), where the subscripts \(\parallel\) and \(\perp\) indicate quantities parallel and perpendicular to the axis of symmetry. Since \({M}_{\parallel}{\sim} {u}_{\parallel}^{2} {\ell}_{\perp}^{2} {\ell}_{\parallel}\) and \({M}_{\perp}{\sim} {u}_{\perp}^{2} {\ell}_{\perp}^{2} {\ell}_{\parallel}\), \(u\) and \(\ell\) being integral scales, self-similarity of the large scales (when it applies) demands \({u}_{\parallel}^{2} {\ell}_{\perp}^{2} {\ell}_{\parallel}= \text{constant}\) and \({u}_{\perp}^{2} {\ell}_{\perp}^{2} {\ell}_{\parallel}= \text{constant}\). This, in turn, requires that \({u}_{\parallel}^{2}/{u}_{\perp}^{2}\) is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that \({M}_{\parallel}\) and \({M}_{\perp}\) are indeed invariants and that, in the fully developed state, \({u}_{\parallel}^{2}/{u}_{\perp}^{2} = \text{constant}\). Somewhat surprisingly, the same simulations also show that \({\ell}_{\parallel}/{\ell}_{\perp}\) is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when \({u}_{\parallel}^{2}/{u}_{\perp}^{2}\) and \({\ell}_{\parallel}/{\ell}_{\perp}\) are both constant, the integral scales should evolve as \({u}_{\perp}^{2}{\sim} {u}_{\parallel}^{2}{\sim} {t}^{-6/5}\) and \({\ell}_{\perp}{\sim} {\ell}_{\parallel}{\sim} {t}^{2/5}\), irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by P. G. Saffman in [Phys. Fluids 10, No. 6, 1349 ff (1967)], are verified by the numerical simulations.