In the study of trustworthy Natural Language Processing (NLP), a number of important research fields have emerged, including that of explainability and privacy. While research interest in both explainable and privacy-preserving NLP has increased considerably in recent years, there remains a lack of investigation at the intersection of the two. This leaves a considerable gap in understanding of whether achieving both explainability and privacy is possible, or whether the two are at odds with each other. In this work, we conduct an empirical investigation into the privacy-explainability trade-off in the context of NLP, guided by the popular overarching methods of Differential Privacy (DP) and Post-hoc Explainability. Our findings include a view into the intricate relationship between privacy and explainability, which is formed by a number of factors, including the nature of the downstream task and choice of the text privatization and explainability method. In this, we highlight the potential for privacy and explainability to co-exist, and we summarize our findings in a collection of practical recommendations for future work at this important intersection.