  This paper attempts to investigate the space of various characteristic
classes for smooth manifold bundles with local system on the total space
inducing a finite holonomy covering. These classes are known as twisted higher
torsion classes. We will give a system of axioms that we require these
cohomology classes to satisfy. Higher Franz Reidemeister torsion and twisted
versions of the higher Miller-Morita-Mumford classes will satisfy these axioms.
We will show that the space oftwisted torsion invariants is two dimensional or
one dimensional depending on the torsion degree and spanned by these two
classes. The proof will greatly depend on results on the equivariant Hatcher
constructions developed in a separate paper.
