We create two-layer interwoven surfaces with (a) planar faces from (b) arbitrary input meshes such that these can be built from cardboard or any other planar material. There are pre-existing constructions for symmetric and regular meshes but these are lacking one or both of the before mentioned attributes (a), (b). We want to emphasize that there do not exist solutions for edge-interpolating or barycentric-apex-created two-layer weavings in general. Hence, we propose constructions in terms of approximations, missing one of the conditions (a), (b) alone, but not both.