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Employing graph neural networks (GNNs) to learn cohesive and discriminative node representations for clustering has shown promising results in deep graph clustering. However, existing methods disregard the reciprocal relationship between representation learning and structure augmentation. This study suggests that enhancing embedding and structure synergistically becomes imperative for GNNs to unleash their potential in deep graph clustering. A reliable structure promotes obtaining more cohesive node representations, while high-quality node representations can guide the augmentation of the structure, enhancing structural reliability in return. Moreover, the generalization ability of existing GNNs-based models is relatively poor. While they perform well on graphs with high homogeneity, they perform poorly on graphs with low homogeneity. To this end, we propose a graph clustering framework named Synergistic Deep Graph Clustering Network (SynC). In our approach, we design a Transform Input Graph Auto-Encoder (TIGAE) to obtain high-quality embeddings for guiding structure augmentation. Then, we re-capture neighborhood representations on the augmented graph to obtain clustering-friendly embeddings and conduct self-supervised clustering. Notably, representation learning and structure augmentation share weights, significantly reducing the number of model parameters. Additionally, we introduce a structure fine-tuning strategy to improve the model's generalization. Extensive experiments on benchmark datasets demonstrate the superiority and effectiveness of our method. The code is released on GitHub and Code Ocean.