Self-assembly of nanoparticles (NPs) inside drying emulsion droplets provides a general strategy for hierarchical structuring of matter at different length scales. The local orientation of neighboring crystalline NPs can be crucial to optimize for instance the optical and electronic properties of the self-assembled superstructures. By integrating experiments and computer simulations, we demonstrate that the orientational correlations of cubic NPs inside drying emulsion droplets are significantly determined by their flat faces. We analyze the rich interplay of positional and orientational order as the particle shape changes from a sharp cube to a rounded cube. Sharp cubes strongly align to form simple-cubic superstructures whereas rounded cubes assemble into icosahedral clusters with additionally strong local orientational correlations. This demonstrates that the interplay between packing, confinement and shape can be utilized to develop new materials with novel properties.