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An $n$-dimensional closed flat manifold is said to be of diagonal type if thestandard representation of its holonomy group $G$ is diagonal. An$n$-dimensional Bieberbach group of diagonal type is the fundamental group ofsuch a manifold. We introduce the diagonal Vasquez invariant of $G$ as theleast integer $n_d(G)$ such that every flat manifold of diagonal type withholonomy $G$ fibers over a flat manifold of dimension at most $n_d(G)$ withflat torus fibers. Using a combinatorial description of Bieberbach groups ofdiagonal type, we give both upper and lower bounds for this invariant. We showthat the lower bounds are exact when $G$ has low rank. We apply this to analysediffuseness properties of Bieberbach groups of diagonal type. This leads to acomplete classification of Bieberbach groups of diagonal type with Kleinfour-group holonomy and to an application to Kaplansky's Unit Conjecture.

Fibering flat manifolds of diagonal type and their fundamental groups