  We introduce a notion of Morita equivalence for non-selfadjoint operator
algebras equipped with a completely isometric involution (operator *-algebras).
We then show that the unbounded Kasparov product by a Morita equivalence
bimodule induces an isomorphism between equivalence classes of twisted spectral
triples over Morita equivalent operator *-algebras. This leads to a tentative
definition of unbounded bivariant K-theory and we prove that this bivariant
theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded
transform. Moreover, the unbounded Kasparov product provides a refinement of
the usual interior Kasparov product. We illustrate our results by proving
C^1-versions of well-known C^*-algebraic Morita equivalences in the context of
hereditary subalgebras, conformal equivalences and crossed products by discrete
groups.
