  The conformal loop ensemble (CLE) is the canonical conformally invariant
probability measure on non-crossing loops in a simply connected domain in
$\mathbb C$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider
CLE$_\kappa$ on the whole-plane in the regime in which the loops are
self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the
inversion map $z \mapsto 1/z$. This shows that whole-plane CLE$_\kappa$ for
$\kappa \in (4,8)$ defines a conformally invariant measure on loops on the
Riemann sphere. The analogous statement in the regime in which the loops are
simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together
with the present work covers the entire range $\kappa \in (8/3,8)$ for which
CLE$_\kappa$ is defined. As an intermediate step in the proof, we show that
CLE$_\kappa$ for $\kappa \in (4,8)$ on an annulus, with any specified number of
inner-boundary-surrounding loops, is well-defined and conformally invariant.
