As an innovative paradigm for the next-generation 6G communication, integrated space and terrestrial networks (ISTNs) combine emerging satellite mega-constellations and existing terrestrial network infrastructures, promising to provide ubiquitous and high-speed Internet services for global users. However, as satellites are operated in the open and uncontrolled space environment, future ISTNs are vulnerable to various attacks. Towards a secure and reliable ISTN for pervasive Internet services globally, we carry out our study in three steps. First, we conduct an in-depth analysis of the unique characteristics and new vulnerabilities of emerging ISTNs. Second, we propose space digital twin (SDT), a novel technology that can facilitate secure and reliable satellite Internet with the ability to conduct multi-dimensional security assessment, identify various vulnerabilities, and evaluate the effectiveness of countermeasures accordingly. With SDT techniques, an ISTN operator/researcher can flexibly build a virtual ISTN environment, which duplicates the network architecture, protocols, behaviors as well as security threats and vulnerabilities of a real ISTN, for security assessment and countermeasures evaluation. Finally, we conclude the key challenges of realizing the full capability of SDT, and accordingly outline a list of corresponding new directions for future research.