Refactoring is the process of restructuring existing code without changing its external behavior while improving its internal structure. Refactoring engines are integral components of modern Integrated Development Environments (IDEs) and can automate or semi-automate this process to enhance code readability, reduce complexity, and improve the maintainability of software products. Similar to traditional software systems such as compilers, refactoring engines may also contain bugs that can lead to unexpected behaviors. In this paper, we propose a novel approach called RETESTER, a LLM-based framework for automated refactoring engine testing. Specifically, by using input program structure templates extracted from historical bug reports and input program characteristics that are error- prone, we design chain-of-thought (CoT) prompts to perform refactoring-preserving transformations. The generated variants are then tested on the latest version of refactoring engines using differential testing. We evaluate RETESTER on two most popular modern refactoring engines (i.e., ECLIPSE, and INTELLIJ IDEA). It successfully revealed 18 new bugs in the latest version of those refactoring engines. By the time we submit our paper, seven of them were confirmed by their developers, and two were fixed.