Metasurfaces present a potent platform to manipulate light by the spatial arrangement of sub-wavelength patterns with well-defined sizes and geometries, in thin films. Metasurfaces by definition are planar. However, it would be highly desirable to integrate metasurfaces with diverse, spatially programmed sub-wavelength features into a 3D monolith, to manipulate light within a compact 3D space. Here, a 3D photoengraving strategy is presented; that is, generation of such composite metasurfaces from a single microstructure via the irradiation of multiple interference laser beams onto different facets of the parent azopolymeric microstructure. Through "photofluidization," this technique enables independent inscription and erasing of metasurfaces onto and from individual facets of 3D monoliths with arbitrary shapes and dimensions, in a high-throughput fashion (over approximately a few cm² at a time). By engraving discrete sub-wavelength 1D surface relief gratings of different pitches on different facets of an inverse pyramidal array, a multiplexing structure-color filter is demonstrated.