  We extend Nekrashevych's $KK$-duality for $C^*$-algebras of regular,
recurrent, contracting self-similar group actions to regular, contracting
self-similar groupoid actions on a graph, removing the recurrence condition
entirely and generalising from a finite alphabet to a finite graph.
  More precisely, given a regular and contracting self-similar groupoid $(G,E)$
acting faithfully on a finite directed graph $E$, we associate two
$C^*$-algebras, $\mathcal{O}(G,E)$ and $\widehat{\mathcal{O}}(G,E)$, to it and
prove that they are strongly Morita equivalent to the stable and unstable
Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the
self-similar limit space. That these algebras are Spanier-Whitehead dual in
$KK$-theory follows from the general result for Ruelle algebras of irreducible
Smale spaces proved by Kaminker, Putnam, and the last author.
