  Twisted spectral triples are a twisting of the notion of spectral triple
aiming at dealing with some type III geometric situations. In the first part of
the paper, we give a geometric construction of the index map of a twisted
spectral triple in terms of $\sigma$-connections on finitely generated
projective modules. This makes it more transparent the analogy with the indices
of Dirac operators with coefficients in vector bundles. In the second part, we
give a direct construction of the Connes-Chern character of a twisted spectral,
both in the invertible and non-invertible cases. Combining these two parts we
obtain an analogue the Atiyah-Singer index formula for twisted spectral
triples.
