  We consider cycles for graded $C^{*,r}$-algebras (Real $C^{*}$-algebras)
which are compatible with the $*$-structure and the real structure. Their
characters are cyclic cocycles. We define a Connes type pairing between such
characters and elements of the van Daele $K$-groups of the $C^{*,r}$-algebra
and its real subalgebra. This pairing vanishes on elements of finite order. We
define a second type of pairing between characters and $K$-group elements which
is derived from a unital inclusion of $C^{*}$-algebras. It is potentially
non-trivial on elements of order two and torsion valued. Such torsion valued
pairings yield topological invariants for insulators. The two-dimensional
Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special
cases of torsion valued pairings. We compute the pairings for a simple class of
periodic models and establish structural results for two dimensional aperiodic
models with odd time reversal invariance.
