Establishing a precise connection between imaged intensity and the incident wavefront is essential for emerging applications in adaptive optics, holography, computational microscopy, and non-line-of-sight imaging. While prior work has shown that breaking symmetries in pupil design enables wavefront recovery from a single intensity measurement, there is little guidance on how to design a pupil that improves wavefront estimation. In this work we introduce a quantitative asymmetry metric to bridge this gap and, through an extensive empirical study and supporting analysis, demonstrate that increasing asymmetry enhances wavefront recoverability. We analyze the trade-offs in pupil design, and the impact on light throughput along with performance in noise. Both large-scale simulations and optical bench experiments are carried out to support our findings.