Industrial processes rely on sensory data for critical decision-making processes. Extracting actionable insights from the collected data calls for an infrastructure that can ensure the trustworthiness of data. To this end, we envision a blockchain-based framework for the Industrial Internet of Things (IIoT) to address the issues of data management and security. Once the data collected from trustworthy sources are recorded in the blockchain, product lifecycle events can be fed into data-driven systems for process monitoring, diagnostics, and optimized control. In this regard, we leverage digital twins (DTs) that can draw intelligent conclusions from the data by identifying the faults and recommending precautionary measures ahead of critical events. Furthermore, we discuss the integration of DTs and blockchain to target key challenges of disparate data repositories, untrustworthy data dissemination, and fault diagnosis. Finally, we identify outstanding challenges faced by the IIoT and future research directions while leveraging blockchain and DTs.