This study is motivated by three-dimensional flows about protrusions and cavities with an arbitrary angle between the external flow and rigid elements. The novel type of a "building block" cavity flow is proposed where the cavity lid moves along its diagonal (Case A). The proposed case is taken as a typical representative of essentially three-dimensional highly separated vortical flows having simple single-block rectangular geometry of computational domain. Computational results are compared to the previous studies where the lid moves parallel to the cavity side walls (Case B). These 3-D lid-driven cavity flows are studied by numerical modeling using second-order upwind schemes for convective terms. The volume and plane integrals of primary and transversal momentum are introduced to compare cases in a quantitative way. For the laminar flow in the cubic cavity, the integral momentum of the secondary flow (which is perpendicular to the lid direction) is about an order of magnitude larger than that in Case B. In Case A, the number of secondary vortices substantially depends on the Re number. The secondary vortices in the central part of the cavity in Case A distinguishes it from Case B, where only corner secondary vortices appear. For a rectangular 3-D 3: 1 : 1 cavity the integral momentum of the secondary flow in Case A is an order of magnitude larger than that in the benchmark cases. The flow field in Case A includes a curvilinear separation line and non-symmetrical vortices which are discussed in the paper. The estimated Goertler number is approximately 4.5 times larger in Case A than that in Case B for the same Re number. This indicates that in Case A the flow becomes unsteady for smaller Re numbers than in Case B. For developed turbulent flow in the cubic cavity, the yaw effect on amplifcation of secondary flow is as strong as that for the laminar flow despite the more complex vortical flow pattern in benchmark case B.