  We investigate the metric and cohomological properties of higher dimensional
analogues of Inoue surfaces, that were introduced by Endo and Pajitnov. We
provide a solvmanifold structure and show that in the diagonalizable case, they
are formal and have invariant de Rham cohomology. Moreover, we obtain an
arithmetic and cohomological characterization of pluriclosed and
astheno-K\"ahler metrics and show they give new examples in all complex
dimensions.
