Vitamin E (alpha-tocopherol) and a range of other biological compounds have long been known to promote the H-II (inverted hexagonal) phase in lipids. Now, it has been well established that purely hydrophobic lipids such as dodecane promote the H-II phase by relieving extensive packing stress. They do so by residing deep within the hydrocarbon core. However, we argue from X-ray diffraction data obtained with 1-palmitoyl-2-oleoylphosphatidylcholine (POPE) and 1,2-dioleoylphosphatidylcholine (DOPE) that alpha-tocopherol promotes the H-II phase by a different mechanism. The OH group on the chromanol moiety of alpha-tocopherol anchors it near the aqueous interface. This restriction combined with the relatively short length of alpha-tocopherol (as compared to DOPE and POPE) means that alpha-tocopherol promotes the H-II phase by relieving compressive packing stress. This observation offers new insight into the nature of packing stress and lipid biophysics. With the deeper understanding of packing stress offered by our results, we also explore the role that molecular structure plays in the primary function of vitamin E, which is to prevent the oxidation of polyunsaturated membrane lipids.