Ensuring trustworthiness in machine learning (ML) models is a multi-dimensional task. In addition to the traditional notion of predictive performance, other notions such as privacy, fairness, robustness to distribution shift, adversarial robustness, interpretability, explainability, and uncertainty quantification are important considerations to evaluate and improve (if deficient). However, these sub-disciplines or 'pillars' of trustworthiness have largely developed independently, which has limited us from understanding their interactions in real-world ML pipelines. In this paper, focusing specifically on compositions of functions arising from the different pillars, we aim to reduce this gap, develop new insights for trustworthy ML, and answer questions such as the following. Does the composition of multiple fairness interventions result in a fairer model compared to a single intervention? How do bias mitigation algorithms for fairness affect local post-hoc explanations? Does a defense algorithm for untargeted adversarial attacks continue to be effective when composed with a privacy transformation? Toward this end, we report initial empirical results and new insights from 9 different compositions of functions (or pipelines) on 7 real-world datasets along two trustworthy dimensions - fairness and explainability. We also report progress, and implementation choices, on an extensible composer tool to encourage the combination of functionalities from multiple pillars. To-date, the tool supports bias mitigation algorithms for fairness and post-hoc explainability methods. We hope this line of work encourages the thoughtful consideration of multiple pillars when attempting to formulate and resolve a trustworthiness problem.