  We examine the index data associated to twisted spectral triples and higher
order spectral triples. In particular, we show that a Lipschitz regular twisted
spectral triple can always be `logarithmically dampened' through functional
calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same
procedure turns higher order spectral triples into spectral triples. We provide
examples of highly regular twisted spectral triples with nontrivial index data
for which Moscovici's ansatz for a twisted local index formula is identically
zero.
