As an indispensable component of the emerging 6G networks, Space-Air-Ground Integrated Networks (SAGINs) are envisioned to provide ubiquitous network connectivity and services by integrating satellite networks, aerial networks, and terrestrial networks. In 6G SAGINs, a wide variety of network services with the features of diverse requirements, complex mobility, and multi-dimensional resources will pose great challenges to service provisioning, which urges the development of service-oriented SAGINs. In this paper, we conduct a comprehensive review of 6G SAGINs from a new perspective of service-oriented network. First, we present the requirements of service-oriented networks, and then propose a service-oriented SAGINs management architecture. Two categories of critical technologies are presented and discussed, i.e., heterogeneous resource orchestration technologies and the cloud-edge synergy technologies, which facilitate the interoperability of different network segments and cooperatively orchestrate heterogeneous resources across different domains, according to the service features and requirements. In addition, the potential future research directions are also presented and discussed.