Metrics on doubles as an inverse semigroup II

We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we give several descriptions of the set $E(M(X))$ of idempotents of this inverse semigroup and its Stone dual space $\widehat{X}$. We also construct $\sigma$-additive measures on $\widehat{X}$ from finitely additive probability measures on $X$ that vanish on bounded subsets.