Trustworthy AI encompasses many aspirational aspects for aligning AI systems with human values, including fairness, privacy, robustness, explainability, and uncertainty quantification. However, efforts to enhance one aspect often introduce unintended trade-offs that negatively impact others, making it challenging to improve all aspects simultaneously. In this position paper, we review notable approaches to these five aspects and systematically consider every pair, detailing the negative interactions that can arise. For example, applying differential privacy to model training can amplify biases in the data, undermining fairness. Drawing on these findings, we take the position that addressing trustworthiness along each axis in isolation is insufficient. Instead, research on Trustworthy AI must account for intersectionality between aspects and adopt a holistic view across all relevant axes at once. To illustrate our perspective, we provide guidance on how researchers can work towards integrated trustworthiness, a case study on how intersectionality applies to the financial industry, and alternative views to our position.