  We begin by presenting a symmetric version of the circle equivariant
T-duality result in a joint work of the second author with Siye Wu, thereby
generalising the results there. We then initiate the study of twisted
equivariant Courant algebroids and equivariant generalised geometry and apply
it to our context. As before, T-duality exchanges type II A and type II B
string theories. In our theory, both spacetime and the T-dual spacetime can be
singular spaces when the fixed point set is non-empty; the singularities
correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges
of type II string theories on the singular spaces are classified by twisted
equivariant cohomology groups, consistent with the previous work of Mathai and
Wu, and prove that they are naturally isomorphic. We also establish the
corresponding isomorphism of twisted equivariant Courant algebroids.
