Intelligent transportation systems (ITS) have been attracting the attention of industry and academia alike for addressing issues raised by the 2030 agenda for sustainable development goals (SDG) approved by the United Nations. However, the diversity and dynamics of present-day transportation scenarios are already very complex, turning the management of ITS into a virtually impossible task for conventional traffic control centers. Recently, the digital twin (DT) paradigm has been presented as a modern architectural concept to tackle complex problems, such as the ones faced by ITS. This survey aims to provide a piece-wise approach to introducing DTs into sustainable ITS by addressing the following cornerstone aspects: i) Why should one consider DTs in ITS applications? ii) What can DTs represent from ITS' new physical environments? And iii) How can one use DTs to address SDG related to efficiency, safety, and ecology in ITS? Our methodological approach for surveying the literature addresses these questions by categorizing contributions and discriminating their ITS elements and agents against the SDG they addressed. Thus, in this survey, we provide an in-depth and contextualized overview of the challenges when approaching ITS through DTs, including scenarios involving autonomous and connected vehicles, ITS' infrastructure, and traffic agents' behavior. Moreover, we propose a functional reference framework for developing DTs of ITS. Finally, we also offer research challenges regarding standardization, connectivity infrastructure, security and privacy aspects, and business management for properly developing DTs for sustainable ITS.