Graph-level clustering, which is essential in medical, biomedical, and social network data analysis, aims to group a set of graphs into various clusters. However, existing methods generally rely on a single clustering criterion, e.g., $k$-means, which limits their abilities to fully exploit the complex Euclidean and structural information inherent in graphs. To bridge this gap, we propose a dual contrastive graph-level clustering (DCGLC) method in this paper. DCGLC leverages graph contrastive learning and introduces the Euclidian-based and subspace-based cluster heads to capture the cluster information from different cluster perspectives. To overcome the inconsistency estimations and fuse the cluster information of multiple cluster heads, we propose a contrastive mechanism to align the cluster information derived from them. The cluster-perspective contrast facilitates the capture of more comprehensive cluster information. Importantly, DCGLC is an end-to-end framework in which graph contrastive learning and cluster-perspective contrast are mutually improved. We demonstrate the superiority of DCGLC over the state-of-the-art baselines on numerous graph benchmarks.