  We present a duality between spectral triples and pseudo-Riemannian spectral
triples. This duality is based on the fundamental connection between twists and
Krein products. A concept of isomorphism of spectral triples is introduced,
transforming one spectral triple to the dual one. In the case of
even-dimensional manifolds, we show how this implements a signature change
through the parity operator, given by the twist itself. The signature change
transformation is then only parametrized by one operator, given by the unitary
that implements the twist. This unitary is a central ingredient of the
approach, being directly connected to the Krein product, the twist and the
parity operator that implement the signature change.
