  We factorize the Dirac operator on the Connes-Landi 4-sphere in unbounded
KK-theory. We show that a family of Dirac operators along the orbits of the
torus action defines an unbounded Kasparov module, while the Dirac operator on
the principal orbit space -an open quadrant in the 2-sphere- defines a
half-closed chain. We show that the tensor sum of these two operators coincides
up to unitary equivalence with the Dirac operator on the Connes-Landi sphere
and prove that this tensor sum is an unbounded representative of the interior
Kasparov product in bivariant K-theory.
  We also generalize our results to Dirac operators on all toric noncommutative
manifolds subject to a condition on the principal stratum. We find that there
is a curvature term that arises as an obstruction for having a tensor sum
decomposition in unbounded KK-theory. This curvature term can however not be
detected at the level of bounded KK-theory.

Operator Algebras;Differential Geometry;Functional Analysis