Our direct numerical simulations show that a coherent structure (scrC) in an initially fine-scale homogeneous, isotropic turbulent field breeds secondary structures in its vicinity. These are organized. Shaped like concentric spiral threads perpendicular to the axis of scrC, each is found to be highly polarized with the azimuthal vorticity component being dominant. The threads occur in pairs, and the polarization typically alternates between adjacent threads. Above a critical value (1000) of ReΓscrC/ν (ΓscrC is circulation, ν is kinematic viscosity) a small number of circulation-rich threads emerge as a result of the evolution. The secondary structures are of both practical and theoretical importance. For Re>Recrit, the strongest threads excite bending waves on the axis of scrC. For ReRecrit, scrC is eventually destroyed. We believe that this feedback phenomenon is of critical importance for the rearrangement of coherent structures (CS) and transition to turbulence in shear flows such as plane, circular, and elliptic jets. Turbulent mixing near scrC is due to entrainment and ejection of fluid by the threads. Local isotropy assumption cannot be applied near a CS, because our results show anisotropy in a layer surrounding scrC. The threads are shown to be the combined result of three mechanisms: (1) azimuthal alignment of small-scale vorticity by the strain rate field of scrC; (2) merger (pairing) and axisymmetrization as in two-dimensional turbulent flows, enabled by the alignment; and (3) polarization by differential rotation.