The Metaverse has emerged to extend our lifestyle beyond physical limitations. As essential components in the Metaverse, digital twins (DTs) are the digital replicas of physical items. End users access the Metaverse using a variety of devices (e.g., head-mounted devices (HMDs)), mostly lightweight. Multi-access edge computing (MEC) and edge networks provide responsive services to the end users, leading to an immersive Metaverse experience. With the anticipation to represent physical objects, end users, and edge computing systems as DTs in the Metaverse, the construction of these DTs and the interplay between them have not been 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). We also study the interplay between the DTs and DTECs to allocate the resources fairly and adequately and provide an immersive experience in the Metaverse. Owing to the dynamic network states (e.g., channel states) and mobility of the users, we discuss the interplay between local DTECs (on local MEC servers) and the global DTEC (on cloud server) to cope with the handover among MEC servers and avoid intermittent Metaverse services.