  We introduce conformal Courant algebroids, a mild generalization of Courant
algebroids in which only a conformal structure rather than a bilinear form is
assumed. We introduce exact conformal Courant algebroids and show they are
classified by pairs $(L,H)$ with $L$ a flat line bundle and $H \in H^3(M,L)$ a
degree 3 class with coefficients in $L$. As a special case gerbes for the
crossed module $({\rm U}(1) \to \mathbb{Z}_2)$ can be used to twist $TM \oplus
T^*M$ into a conformal Courant algebroid. In the exact case there is a twisted
cohomology which is 4-periodic if $L^2 = 1$. The structure of Conformal Courant
algebroids on circle bundles leads us to construct a T-duality for orientifolds
with free involution. This incarnation of T-duality yields an isomorphism of
4-periodic twisted cohomology. We conjecture that the isomorphism extends to an
isomorphism in twisted $KR$-theory and give some calculations to support this
claim.
