Abstract: Graph-to-text (G2T) generation and text-to-graph (T2G) triple extraction are two essential tasks for knowledge graphs. Existing unsupervised approaches become suitable candidates for jointly learning the two tasks due to their avoidance of using graph-text parallel data. However, they adopt multiple complex modules and still require entity information or relation type for training. To this end, we propose INFINITY, a simple yet effective unsupervised method with a unified pretrained language model that does not introduce external annotation tools or additional parallel information. It achieves fully unsupervised graph-text mutual conversion for the first time. Specifically, INFINITY treats both G2T and T2G as a bidirectional sequence generation task by fine-tuning only one pretrained seq2seq model. A novel back-translation-based framework is then designed to generate synthetic parallel data automatically. Besides, we investigate the impact of graph linearization and introduce the structure-aware fine-tuning strategy to alleviate possible performance deterioration via retaining structural information in graph sequences. As a fully unsupervised framework, INFINITY is empirically verified to outperform state-of-the-art baselines for G2T and T2G tasks. Additionally, we also devise a new training setting called cross learning for low-resource unsupervised information extraction.