X-ray tensor tomography system (34,134), comprising - a source for providing a beam with coherent X-rays along a beam path, - an interferometer arranged along the beam path downstream the source comprising a first grating (16, 116) with first grating lines and a second grating (18, 118) with second grating lines, wherein - the second grating lines are parallel to the first grating lines, - the XTT-system (34, 134) is configured to relatively shift the first grating (16, 116) and the second grating (18, 118) with respect to each other in a shifting direction into different grating positions, wherein the shifting direction is parallel to the planes of the gratings (16, 116; 18, 118), - a detector arranged downstream the source and the interferometer for acquiring a sequence of two-dimensional intensity distributions of X-rays of the beam, characterised by - a stage for mounting a specimen in the beam path downstream the source, wherein the stage is designed to rotate the specimen about at least one first axis of rotation and about not more than two axes of rotation (24, 26) for orienting the specimen in the beam path and with respect to the beam path, wherein the first axis of rotation lies in a plane which is tilted - by an angle ψ with respect to the to the planes of the gratings (16, 18), wherein o° < ψ ≤ 90°, and - by an angle β with respect to a plane being orthogonal to the direction of the beam path at a location of the stage, wherein o° < β ≤ 90°, - a reconstruction unit configured to reconstruct scattering tensors for a specimen containing one or more structures capable of scattering X-rays, wherein the scattering tensors provide information about the three-dimensional position of the one or more structures and about an orientation of the one or more structures, based on - a sequence of two-dimensional intensity distributions acquired for different orientations of the specimen and for different grating positions, and on - the orientations and grating positions, for which the sequence of two-dimensional intensity distributions is acquired. - a source for providing a beam with coherent X-rays along a beam path, - an interferometer arranged along the beam path downstream the source comprising a first grating (16, 116) with first grating lines and a second grating (18, 118) with second grating lines, wherein - the second grating lines are parallel to the first grating lines, - the XTT-system (34, 134) is configured to relatively shift the first grating (16, 116) and the second grating (18, 118) with respect to each other in a shifting direction into different grating positions, wherein the shifting direction is parallel to the planes of the gratings (16, 116; 18, 118), - the second grating lines are parallel to the first grating lines, - the XTT-system (34, 134) is configured to relatively shift the first grating (16, 116) and the second grating (18, 118) with respect to each other in a shifting direction into different grating positions, wherein the shifting direction is parallel to the planes of the gratings (16, 116; 18, 118), - a detector arranged downstream the source and the interferometer for acquiring a sequence of two-dimensional intensity distributions of X-rays of the beam, characterised by - a stage for mounting a specimen in the beam path downstream the source, wherein the stage is designed to rotate the specimen about at least one first axis of rotation and about not more than two axes of rotation (24, 26) for orienting the specimen in the beam path and with respect to the beam path, wherein the first axis of rotation lies in a plane which is tilted - by an angle ψ with respect to the to the planes of the gratings (16, 18), wherein o° < ψ ≤ 90°, and - by an angle β with respect to a plane being orthogonal to the direction of the beam path at a location of the stage, wherein o° < β ≤ 90°, - by an angle ψ with respect to the to the planes of the gratings (16, 18), wherein o° < ψ ≤ 90°, and - by an angle β with respect to a plane being orthogonal to the direction of the beam path at a location of the stage, wherein o° < β ≤ 90°, - a reconstruction unit configured to reconstruct scattering tensors for a specimen containing one or more structures capable of scattering X-rays, wherein the scattering tensors provide information about the three-dimensional position of the one or more structures and about an orientation of the one or more structures, based on - a sequence of two-dimensional intensity distributions acquired for different orientations of the specimen and for different grating positions, and on - the orientations and grating positions, for which the sequence of two-dimensional intensity distributions is acquired. - a sequence of two-dimensional intensity distributions acquired for different orientations of the specimen and for different grating positions, and on - the orientations and grating positions, for which the sequence of two-dimensional intensity distributions is acquired.