An equilibrium similarity analysis is applied to the transport equation for <(δ q)2> (≡ <(δ u 2 + <(δ v)2> + <(δ w 2>), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy decays with a power-law behaviour ( ∼ xm), and the characteristics length scale, which is readily identifiable with the Taylor microscale, varies as x1/2. This solution is identical to that obtained by George (1992) from the spectral energy equation. The solution does not depend on the actual magnitude of the Taylor-microscale Reynolds number Rλ (∼ 1/2 λ/v); Rλ should decay as x(m+1)/2 when m < -1. The solution is tested at relatively low Rλ against grid turbulence data for which m ≃ -1.25 and Rλ decays as x-0.125. Although homogeneity and isotropy are poorly approximated in this flow, the measurements of <(δ q 2> and, to a lesser extent, <(δ u)(δ q 2>, satisfy similarly reasonably over a significant range of r/λ, where r is the streamwise separation across which velocity increments are estimated. For this range, a similarity-based calculation of the third-order structure function <(δ u (δ q)2> is in reasonable agreement with measurements. Kolmogorov-normalized distributions of <(δ q)2> and <(δ u)(δ q)2> collapse only at small r. Assuming homogeneity, isotropy and a Batchelor-type parameterization for <(δ q)2>, it is found that Rλ may need to be as large as 106 before a two-decade inertial range is observed.