The nanostructure gradient along the fiber radius inside polymer strands is uncovered by smallangle X-ray scattering (SAXS) microbeam-scanning experiments and X-ray scattering fiber computer-tomography (XSF-CT) analysis. We notice restricted visibility of scattering features within the series of patterns. The reason is violation of local fiber symmetry (LFS) in the irradiated volume elements (voxels). For its theoretical treatment, a set of elementary topologies (tangential grain, radial grain) is introduced. Systematic aberrations (ultrareconstruction, infrareconstruction) generated by tomographic reconstruction of affected series are described. A concept for handling and utilization of these aberrations for nanostructure analysis is devised. Precursors of polymer microfibrillar-reinforced composites (MFC) containing poly(ether)-block-amide (PEBA) and poly(ethylene terephthalate) (PET) with varying cold-draw ratio are studied. We compare results from a direct analysis of the smeared measured patterns to results obtained after tomographic reconstruction and fathom the power of reconstruction methods. Ideas for advanced practical applications of the XSF-CT method are discussed.