  We introduce a pair of isospectral but non-isometric compact flat 3-manifolds
called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra
and Didi are very different. Where Tetra has two quarter-twisting geodesics of
the shortest length, Didi has four half-twisting geodesics. Nevertheless, these
spaces are isospectral. This isospectrality can be proven directly by matching
eigenfunctions having the same eigenvalue. However, the real interest of this
pair--and what led us to discover it--is the way isospectrality emerges from
the Selberg trace formula, as the result of a delicate interplay between the
lengths and twists of closed geodesics.
