  We characterize Carnot groups admitting a 1-quasiconformal metric inversion
as the Lie groups of Heisenberg type whose Lie algebras satisfy the
$J^2$-condition, thus characterizing a special case of inversion invariant
bi-Lipschitz homogeneity. A more general characterization of inversion
invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is
also provided.
