The distribution of three dimensional intrinsic galaxy shapes has been a longstanding open question. The difficulty stems from projection effects meaning one must rely on statistical methods applied to galaxy samples to infer intrinsic shape distributions. Theoretical work using analytical galaxy potentials suggests a relationship between galaxy intrinsic shape (as defined by its triaxiality, in practice a proxy for how prolate a galaxy is) and the intrinsic misalignment angle between kinematic and morphological axes ($Psi_{rm int}$). This relationship reduces the number of unknowns, providing more reliable inferred intrinsic shape distributions than methods using photometry alone. Here we explore the connection between intrinsic shape and stellar kinematics using cosmological hydrodynamical simulations from the Illustris project. The strongest relationship we find is that galaxy intrinsic flattening is correlated with specific angular momentum (j) with high j galaxies being flatter than galaxies with low specific angular momentum. Our analysis shows that, although the majority of kinematically misaligned galaxies exhibit prolate shapes, examples of kinematically aligned prolate galaxies are also present. Clearly a direct correspondence between prolate shape and minor-axis rotation (often referred to as prolate rotation) is not present in Illustris. Thus, we demonstrate that the assumption of a simple relationship between $Psi_{rm int}$ and intrinsic shape commonly employed in shape recovery studies is not valid for Illustris galaxies. We suggest improvements on the method as well as some alternative methods for future work in this area.
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