  We present a definition of indefinite Kasparov modules, a generalisation of
unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g.
hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov
module we can associate a pair of (genuine) Kasparov modules, and that this
process is reversible. We present three examples of our framework: the Dirac
operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an
indefinite metric), the harmonic oscillator, and the construction via the
Kasparov product of an indefinite spectral triple from a family of spectral
triples. This last construction corresponds to a foliation of a globally
hyperbolic spacetime by spacelike hypersurfaces.
