  We investigate the structure of circle actions with the Rokhlin property,
particularly in relation to equivariant $KK$-theory. Our main results are
$\mathbb{T}$-equivariant versions of celebrated results of Kirchberg: any
Rokhlin action on a separable, nuclear C*-algebra is $KK^\mathbb{T}$-equivalent
to a Rokhlin action on a Kirchberg algebra; any Rokhlin action on an exact
separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique
Rokhlin action); and two circle actions with the Rokhlin property on a
Kirchberg algebra are conjugate if and only if they are
$KK^\mathbb{T}$-equivalent.
  In the presence of the UCT, $KK^\mathbb{T}$-equivalence for Rokhlin actions
reduces to isomorphism of a $K$-theoretical invariant, namely of a canonical
pure extension naturally associated to any Rokhlin action, and we provide a
complete description of the extensions that arise from actions on nuclear
C*-algebras. In contrast with the non-equivariant setting, an isomorphism
between the $K^\mathbb{T}$-theories of Rokhlin actions on Kirchberg algebras
does not necessarily lift to a $KK^\mathbb{T}$-equivalence.
