Symmetric noncrossing partitions of an annulus with double points

For affine Coxeter groups of affine types $\tilde{D}$ and $\tilde{B}$, we model the interval $[1,c{]}_T$ in the absolute order by symmetric noncrossing partitions of an annulus with one or two double points. In type $\tilde{B}$ (and \emph{almost} in type $\tilde{D}$), the diagrams also model the larger lattice defined by McCammond and Sulway.