Engineering textiles are used as fibrous reinforcements in high performance polymer composites. The mechanical properties of composite materials depend on their dual-scale porous structure: long and elongated microscopic open spaces (micropores) appear between the filaments of fiber tows, and up to two orders of magnitude larger mesoscopic spaces (mesopores) exist between yarns. Because of the complex structure of composites, it is difficult to establish a connection between fiber architecture and mechanical behavior. To achieve this goal, X-ray microtomography (Micro-CT) can be used to provide detailed information on the geometric mesostructure of continuous fiber composites. The objective of this paper is to present a new approach based on “Micro-CT Aided Geometric Modeling” (Micro-CT AGM) to construct detailed geometric models of engineering textiles from Micro-CT three-dimensional (3D) images. The obtained geometric models are called “material twins” because of three distinctive features: (1) they are representative of the material variability; (2) their accuracy can be evaluated; and (3) they can be used to carry out computer simulations of mechanical behavior. This new method is applied to two industrial reinforcements: a 2D plain woven fabric and a 3D orthogonal glass textile. By introducing a new multiple factor morphological accuracy criterion, the quality of the “material twin” reconstruction can be compared to models obtained by standard image processing techniques or textile modeling software.