The present paper concerns the design of the semantic infrastructure of the data space for cultural heritage as envisaged by the European Commission in its recent documents. Due to the complexity of the cultural heritage data and of their intrinsic inter-relationships, it is necessary to introduce a novel ontology, yet compliant with existing standards and interoperable with previous platforms used in this context as Europeana. The data space organization must be tailored to the methods and the theory of cultural heritage, briefly summarized in the introduction. The new ontology is based on the Digital Twin concept, i.e., the digital counterpart of cultural heritage assets incorporating all the digital information pertaining to them. This creates a Knowledge Base on the cultural heritage data space. The paper outlines the main features of the proposed Heritage Digital Twin ontology and provides some examples of its application. Future work will include completing the ontology in all its details and testing it in other real cases and with the various sectors of the cultural heritage community.