Vortex beams with ultra-high brilliance can greatly enrich the light and matter interaction process and even shed light on the unexpected information in relativistic nonlinear optics. Thus, we propose a scheme for relativistic intense vortex harmonic radiation by use of bi-circular Laguerre-Gaussian lasers irradiating relativistic plasmas. According to the law of conservation of photon energy and angular momentum during the generation of higher-order harmonics, the emitted harmonics own controllable spin and orbital angular momentum simultaneously, as the three-dimensional particle-in-cell simulation results shown. Based on this discussion, the methods of adjusting harmonic order, polarization state (spin angular momentum) and topological charge number (orbital angular momentum) are proposed. It is found that if frequency ratio and circular polarization state of the bi-chromatic Laguerre-Gaussian laser are changed, such as with the same right-handed or left-handed circular polarization. The harmonic characteristics, including the harmonic order, the polarization state and the vortex order are flexibly controlled. Therefore, this work provides an efficient and practical approach to produce bright, spectral tunable harmonic radiation with designable spin and orbital angular momentum, which may own the application prospect going from optical communications, bio-photonics, optical micromanipulations to ion accelerations.