  In two previous papers we have computed the inelasticity $\epsilon$ in a
head-on collision of two $D$-dimensional Aichelburg-Sexl shock waves, using
perturbation theory to calculate the geometry in the future light-cone of the
collision. The first order result, obtained as an accurate numerical fit,
yielded the remarkably simple formula $\epsilon_{\rm 1st\, order} = 1/2 - 1/D$.
Here we show, analytically, that this result is exact in first order
perturbation theory. Moreover, we clarify the relation between perturbation
theory and an angular series of the inelasticity's angular power around the
symmetry axis of the collision $(\theta = 0,\pi)$. To establish these results,
firstly, we show that at null infinity the angular dependence factorises order
by order in perturbation theory, as a result of a hidden symmetry. Secondly, we
show that a consistent truncation of the angular series in powers of $\sin^2
\theta$ at some order $O(n)$ requires knowledge of the metric perturbations up
to $O(n+1)$. In particular, this justifies the isotropy assumption used in
first order perturbation theory. We then compute, analytically, all terms that
contribute to the inelasticity and depend linearly on the initial conditions
(surface terms), including second order contributions.
