Attributed graph clustering is a basic yet essential method for graph data exploration. Recent efforts over graph contrastive learning have achieved impressive clustering performance. However, we observe that the commonly adopted InfoMax operation tends to capture redundant information, limiting the downstream clustering performance. To this end, we develop a novel method termed attributed graph clustering with dual redundancy reduction (AGC-DRR) to reduce the information redundancy in both input space and latent feature space. Specifically, for the input space redundancy reduction, we introduce an adversarial learning mechanism to adaptively learn a redundant edge-dropping matrix to ensure the diversity of the compared sample pairs. To reduce the redundancy in the latent space, we force the correlation matrix of the cross-augmentation sample embedding to approximate an identity matrix. Consequently, the learned network is forced to be robust against perturbation while discriminative against different samples. Extensive experiments have demonstrated that AGC-DRR outperforms the state-of-the-art clustering methods on most of our benchmarks. The corresponding code is available at https://github.com/gongleii/AGC-DRR.