Industry 4.0 is having an increasingly positive impact on the value chain by modernizing and optimizing the production and distribution processes. In this streamline, the digital twin (DT) is one of the most cutting-edge technologies of Industry 4.0, providing simulation capabilities to forecast, optimize and estimate states and configurations. In turn, these technological capabilities are encouraging industrial stakeholders to invest in the new paradigm, though an increased focus on the risks involved is really needed. More precisely, the deployment of a DT is based on the composition of technologies such as cyber-physical systems, the Industrial Internet of Things, edge computing, virtualization infrastructures, artificial intelligence and big data. However, the confluence of all these technologies and the implicit interaction with the physical counterpart of the DT in the real world generate multiple security threats that have not yet been sufficiently studied. In that context, this paper analyzes the current state of the DT paradigm and classifies the potential threats associated with it, taking into consideration its functionality layers and the operational requirements in order to achieve a more complete and useful classification. We also provide a preliminary set of security recommendations and approaches that can help to ensure the appropriate and trustworthy use of a DT.