We study the class of vacuum (Ricci flat) six-dimensional spacetimes admitting a non-degenerate multiple Weyl aligned null direction \(\ell\), thus being of Weyl type II or more special. Subject to an additional assumption on the asymptotic fall-off of the Weyl tensor, we prove that these spacetimes can be completely classified in terms of the two eigenvalues of the (asymptotic) twist matrix of \(\ell\) and of a discrete parameter \(U^0=\pm 1/2, 0\). All solutions turn out to be Kerr-Schild spacetimes of type D and they reduce to a family of "generalized" Myers-Perry metrics (which include limits and analytic continuations of the original Myers-Perry metric, such as certain NUT spacetimes). A special subcase corresponds to twisting solutions with zero shear.