A double-sided fabric having elastic properties in its warp direction and provided with gripping properties, formed by the weaving of warp yarns and of weft yarns, and wherein the warp yarns are of four different types worked in different weaves with respect to the weft yarns, respectively: - elastic yarns worked in a taffeta weave; - core yarns, formed of synthetic yarns texturized by false twist on the face, intended to form the female element of the hook-and-loop structure, worked in a plain weave alternating with surface floats and having a linear strand density in the range from 10 to 25 dtex, in addition to a hot air shrinkage rate according to the DIN EN 14621 standard of at least 5%; a crimp contraction greater than 35%, said crimp contraction representing the length ratio between a yarn section submitted to a 2 cN/tex force and the length of this same yarn section with no tension, a crimp modulus greater than 20%, said crimp modulus being the elongation capacity of the texturized yarns in the crimp elasticity range, and a crimp stability according to the DIN EN ISO 53840-T1 standard greater than 65%, - core yarns on the back side, formed of multifilament synthetic yarns on the back, having a linear strand density in the range from 0.3 to 1.7 dtex worked in a rep weave, alternating floats of great length on the back side and short binding areas on the face side, and intended to ensure the comfort and the soft hand of this side of the fabric, - yarns used as binding yarns, exclusively formed of thermo-adhesive synthetic yarns, having a melting point in the range from 80°C to 160°C, worked in a plain weave, ensuring the maintaining of the extensible yarns in the central portion of the fabric and ensuring, after the finishing treatment, the blocking of the core yarns (face yarns), forming the female element of the hook-and-loop system, and the blocking of the core yarns (back yarns). - elastic yarns worked in a taffeta weave; - core yarns, formed of synthetic yarns texturized by false twist on the face, intended to form the female element of the hook-and-loop structure, worked in a plain weave alternating with surface floats and having a linear strand density in the range from 10 to 25 dtex, in addition to a hot air shrinkage rate according to the DIN EN 14621 standard of at least 5%; a crimp contraction greater than 35%, said crimp contraction representing the length ratio between a yarn section submitted to a 2 cN/tex force and the length of this same yarn section with no tension, a crimp modulus greater than 20%, said crimp modulus being the elongation capacity of the texturized yarns in the crimp elasticity range, and a crimp stability according to the DIN EN ISO 53840-T1 standard greater than 65%, - core yarns on the back side, formed of multifilament synthetic yarns on the back, having a linear strand density in the range from 0.3 to 1.7 dtex worked in a rep weave, alternating floats of great length on the back side and short binding areas on the face side, and intended to ensure the comfort and the soft hand of this side of the fabric, - yarns used as binding yarns, exclusively formed of thermo-adhesive synthetic yarns, having a melting point in the range from 80°C to 160°C, worked in a plain weave, ensuring the maintaining of the extensible yarns in the central portion of the fabric and ensuring, after the finishing treatment, the blocking of the core yarns (face yarns), forming the female element of the hook-and-loop system, and the blocking of the core yarns (back yarns).