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Self-supervised entity alignment (EA) aims to link equivalent entities across different knowledge graphs (KGs) without seed alignments. The current SOTA self-supervised EA method draws inspiration from contrastive learning, originally designed in computer vision based on instance discrimination and contrastive loss, and suffers from two shortcomings. Firstly, it puts unidirectional emphasis on pushing sampled negative entities far away rather than pulling positively aligned pairs close, as is done in the well-established supervised EA. Secondly, KGs contain rich side information (e.g., entity description), and how to effectively leverage those information has not been adequately investigated in self-supervised EA. In this paper, we propose an interactive contrastive learning model for self-supervised EA. The model encodes not only structures and semantics of entities (including entity name, entity description, and entity neighborhood), but also conducts cross-KG contrastive learning by building pseudo-aligned entity pairs. Experimental results show that our approach outperforms previous best self-supervised results by a large margin (over 9% average improvement) and performs on par with previous SOTA supervised counterparts, demonstrating the effectiveness of the interactive contrastive learning for self-supervised EA.