The generation of quad meshes based on surface parametrization techniques has proven to be a versatile approach. These techniques quantize an initial seamless parametrization so as to obtain an integer grid map implying a pure quad mesh. State-of-the-art methods following this approach have to assume that the surface to be meshed either has no boundary, or has a boundary which the resulting mesh is supposed to be aligned to. In a variety of applications this is not desirable and non-boundary-aligned meshes or grid-parametrizations are preferred. We thus present a technique to robustly generate integer grid maps which are either boundary-aligned, non-boundary-aligned, or partially boundary-aligned, just as required by different applications. We thereby generalize previous work to this broader setting. This enables the reliable generation of trimmed quad meshes with partial elements along the boundary, preferable in various scenarios, from tiled texturing over design and modeling to fabrication and architecture, due to fewer constraints and hence higher overall mesh quality and other benefits in terms of aesthetics and flexibility.