We show that the gaining component in interacting binaries can rotate faster than its orbital revolution as a consequence of the accretion process. We derive an approximative analytical formula for the Roche lobe radius of asynchronously rotating accretors. We present the case of the semidetached interacting binary TX UMa, for which we measured directly asynchronous rotation of its accretor. We suggest a method to detect indirectly a fast spinning of accretors in symbiotic binaries based on the the X-ray luminosity of the boundary layer. We demonstrate this possibility for the case of EG And.