T-duality: Topology change from H-flux

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E₈ and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over ℝPⁿ, and the Ad S⁵ × S⁵ to Ad S⁵ × ℂP² × S¹ with background H-flux of Duff, Lü 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 G₄ receives a correction. 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.