  We review the Reidemeister torsion, Ray-Singer's analytic torsion and the
Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham
complex twisted by a flux form introduced by the current authors and recall its
properties. We define a new twisted analytic torsion for the complex of
invariant differential forms on the total space of a principal circle bundle
twisted by an invariant flux form. We show that when the dimension is even,
such a torsion is invariant under certain deformation of the metric and the
flux form. Under T-duality which exchanges the topology of the bundle and the
flux form and the radius of the circular fiber with its inverse, the twisted
torsions of invariant forms are inverse to each other for any dimension.
