The rapid expansion of AI-generated content (AIGC) reflects the iteration from assistive AI towards generative AI (GAI) with creativity. Meanwhile, the 6G networks will also evolve from the Internet-of-everything to the Internet-of-intelligence with hybrid heterogeneous network architectures. In the future, the interplay between GAI and the 6G will lead to new opportunities, where GAI can learn the knowledge of personalized data from the massive connected 6G end devices, while GAI's powerful generation ability can provide advanced network solutions for 6G network and provide 6G end devices with various AIGC services. However, they seem to be an odd couple, due to the contradiction of data and resources. To achieve a better-coordinated interplay between GAI and 6G, the GAI-native networks (GainNet), a GAI-oriented collaborative cloud-edge-end intelligence framework, is proposed in this paper. By deeply integrating GAI with 6G network design, GainNet realizes the positive closed-loop knowledge flow and sustainable-evolution GAI model optimization. On this basis, the GAI-oriented generic resource orchestration mechanism with integrated sensing, communication, and computing (GaiRom-ISCC) is proposed to guarantee the efficient operation of GainNet. Two simple case studies demonstrate the effectiveness and robustness of the proposed schemes. Finally, we envision the key challenges and future directions concerning the interplay between GAI models and 6G networks.