Space-air-ground integrated network (SAGIN) is considered as a core requirement in emerging 6G networks, which integrates the terrestrial and non-terrestrial networks to reach the full network coverage and ubiquitous services. To envision the ubiquitous intelligence and the deep integration in 6G SAGIN, a paradigm of cybertwin-enabled 6G SAGIN is presented in this paper. Specifically, a cybertwin-enabled SAGIN architecture is first presented, where a novel five-dimension digital twin (DT) model is presented. Particularly, three categories of critical technologies are presented based on the cybertwin of SAGIN, i.e., cybertwinbased multi-source heterogeneous network integration, cybertwin-based integrated cloud-edge-end, and cybertwin-based integrated sensing-communication-computing. Besides, two open issues in the cybertwin-enabled SAGIN are studied, i.e., the networking decision and optimization and the cybertwin-enabled cross-layer privacy and security, where the challenges are discussed and the potential solutions are directed. In addition, a case study with federal learning is developed and open research issues are discussed.