The Metaverse has emerged to extend our lifestyle beyond physical limitations. As essential components in the Metaverse, digital twins (DTs) are the real-time digital replicas of physical items. Multi-access edge computing (MEC) provides responsive services to the end users, ensuring an immersive and interactive Metaverse experience. While the digital representation (DT) of physical objects, end users, and edge computing systems is crucial in the Metaverse, the construction of these DTs and the interplay between them have not been well-investigated. In this paper, we discuss the bidirectional reliance between the DT and the MEC system and investigate the creation of DTs of objects and users on the MEC servers and DT-assisted edge computing (DTEC). To ensure seamless handover among MEC servers and to avoid intermittent Metaverse services, we also explore the interaction between local DTECs on local MEC servers and the global DTEC on the cloud server due to the dynamic nature of network states (e.g., channel state and users' mobility). We investigate a continual learning framework for resource allocation strategy in local DTEC through a case study. Our strategy mitigates the desynchronization between physical-digital twins, ensures higher learning outcomes, and provides a satisfactory Metaverse experience.