Syntax-controlled paraphrase generation aims to produce paraphrase conform to given syntactic patterns. To address this task, recent works have started to use parse trees (or syntactic templates) to guide generation.A constituency parse tree contains abundant structural information, such as parent-child relation, sibling relation, and the alignment relation between words and nodes. Previous works have only utilized parent-child and alignment relations, which may affect the generation quality. To address this limitation, we propose a Structural Information-augmented Syntax-Controlled Paraphrasing (SI-SCP) model. Particularly, we design a syntax encoder based on tree-transformer to capture parent-child and sibling relations. To model the alignment relation between words and nodes, we propose an attention regularization objective, which makes the decoder accurately select corresponding syntax nodes to guide the generation of words. Experiments show that SI-SCP achieves state-of-the-art performances in terms of semantic and syntactic quality on two popular benchmark datasets. Additionally, we propose a Syntactic Template Retriever (STR) to retrieve compatible syntactic structures. We validate that STR is capable of retrieving compatible syntactic structures. We further demonstrate the effectiveness of SI-SCP to generate diverse paraphrases with retrieved syntactic structures.