There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.

The six orientable ones are determined by their holonomy groups $$ C_1,C_2,C_3,C_4,C_6 $$ and $$ C_2 \times C_2 $$ The five with cyclic holonomy all arise as the mapping torus of a mapping class of $ T^2 $ with the corresponding order: 1,2,3,4, or 6. These five Euclidean manifolds with cyclic holonomy can even be constructed as a quotient of the special Euclidean group $ SE_2 $ by a cocompact lattice constructed as the semidirect product of a lattice in $ \mathbb{R}^2 $ and a finite cyclic subgroup of $ SL_2(\mathbb{Z}) $ preserving that lattice. For example $ C_1 $ corresponds to the three torus $ T^3 $.

**I am very curious about the compact flat orientable 3 manifold with holonomy $ C_2 \times C_2 $ (known as the Hantzsche-Wendt manifold). It is not a mapping torus of $ T^2 $ like the other five, but perhaps it is a mapping torus of the Klein bottle $ K $?**