Extensive prior work has shown that colloidal inorganic nanocrystals coated with organic ligand shells can behave as artificial atoms and, as such, form superlattices with different crystal structures and packing densities. Although ordered superlattices present a high degree of long-range positional order, the relative crystallographic orientation of the inorganic nanocrystals with respect to each other tends to be random. Recent works have shown that superlattices can achieve orientational alignment through combinations of nanocrystal faceting and ligand modification, as well as selective metal particle attachment to particular facets. These studies have focused on the assembly of high-symmetry nanocrystals, such as cubes and cuboctahedra. Here, we study the assembly of elliptically faceted CdSe/CdS core/shell nanocrystals with one distinctive crystallographic orientation along the major elliptical axis. We show that the nanocrystals form an unexpectedly well-ordered translational superlattice, with a degree of order comparable to that achieved with higher-symmetry nanocrystals. Additionally, we show that, due to the particles' faceted shape, the superlattice is characterized by an orientational glass phase in which only certain orientations are possible due to entropically frustrated crystallization. In this phase, the nanocrystals do not exhibit a local orientational ordering but rather have distinct orientations that emerge at different locations within the same domain. The distinct orientations are a result of a facet-to-facet lock-in mechanism that occurs during the self-assembly process. These facet-to-facet alignments force the nanocrystals to tilt on different lattice planes forming different projections that we termed apparent polydispersity. Our experimental realization of an orientational glass phase for multifaceted semiconducting nanocrystals can be used to investigate how this phase is formed and how it can be utilized for potential optical, electrical, and thermal transport applications.