  T-duality acts on circle bundles by exchanging the first Chern class with the
fiberwise integral of the H-flux, as we motivate using E_8 and also using
S-duality. We present known and new examples including NS5-branes,
nilmanifolds, Lens spaces, both circle bundles over RP^n, and the AdS^5 x S^5
to AdS^5 x CP^2 x S^1 with background H-flux of Duff, Lu and Pope. When
T-duality leads to M-theory on a non-spin manifold the gravitino partition
function continues to exist due to the background flux, however the known
quantization condition for G_4 fails. In a more general context, we use
correspondence spaces to implement isomorphisms on the twisted K-theories and
twisted cohomology theories and to study the corresponding
Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable
twists, both twisted theories admit fusion products and so are naturally rings.
