When used in combination with raster scanning, small-angle X-ray scattering (SAXS) has proven to be a valuable imaging technique of the nanoscale(1), for example of bone, teeth and brain matter(2-5). Although two-dimensional projection imaging has been used to characterize various materials successfully, its three-dimensional extension, SAXS computed tomography, poses substantial challenges, which have yet to be overcome. Previous work(6-11) using SAXS computed tomography was unable to preserve oriented SAXS signals during reconstruction. Here we present a solution to this problem and obtain a complete SAXS computed tomography, which preserves oriented scattering information. By introducing virtual tomography axes, we take advantage of the two-dimensional SAXS information recorded on an area detector and use it to reconstruct the full three-dimensional scattering distribution in reciprocal space for each voxel of the three-dimensional object in real space. The presented method could be of interest for a combined six-dimensional real and reciprocal space characterization of mesoscopic materials with hierarchically structured features with length scales ranging from a few nanometres to a few millimetres-for example, biomaterials such as bone or teeth, or functional materials such as fuel-cell or battery components.