  Recently Baraglia showed how topological T-duality can be extended to apply
not only to principal circle bundles, but also to non-principal circle bundles.
We show that his results can also be recovered via two other methods: the
homotopy-theoretic approach of Bunke and Schick, and the noncommutative
geometry approach which we previously used for principal torus bundles. This
work has several interesting byproducts, including a study of the K-theory of
crossed products by Isom(R), the universal cover of O(2), and some interesting
facts about equivariant K-theory for Z/2. In the final section of this paper,
these results are extended to the case of bundles with singular fibers, or in
other words, non-free O(2)-actions.
