The present study aimed to examine whether Mandarin-speaking children on the autism spectrum showed differences in comprehending spatial demonstratives ("this" and "that", and "here" and "there"), as compared to typically developing (TD) children. Another aim of this study was to investigate the roles of theory of mind (ToM) and executive functions (EF) in the comprehension of spatial demonstratives. Twenty-seven autistic children (mean age 6.86) and 27 receptive-vocabulary-matched TD children (mean age 5.82) were recruited. Demonstrative comprehension was assessed based on participants' ability to place objects in certain locations according to experimenters' instructions which involved these demonstratives in three different conditions (same-, opposite-, and spectator-perspective conditions). Four false-belief tasks were administered to measure ToM, and the word-span task and the dimensional change card sort task were used to measure two subcomponents of EF -- working memory and mental flexibility -- respectively. Children on the autism spectrum were found to score below TD children in the comprehension of spatial demonstratives. In addition, the results showed that ToM and working memory were conducive to the correct interpretation of spatial demonstratives. The two cognitive abilities mutually influenced their respective roles in spatial demonstrative comprehension in the three different conditions. The findings suggest that the comprehension of spatial demonstratives comprehension is an area of need in Mandarin-speaking children on the autism spectrum, and it might be linked to their differences in cognitive abilities.