Homochirality, the single-handedness of optically asymmetric chemical structures, is present in all major biological macromolecules. Terrestrial life's preference for one isomer over its mirror image in D-sugars and L-amino acids has both fascinated and puzzled biochemists for over a century. But the contrasting case of the equally fundamental phospholipids has received less attention. Although the phospholipid glycerol headgroups of archaea and bacteria are both exclusively homochiral, the stereochemistries between the two domains are opposite. Here I argue that the reason for this "dual homochirality" was a simple evolutionary choice at the independent origin of the two synthesizing enzymes. More broadly, this points to a trivial biogenic cause for the evolution of homochirality: the enzymatic processes that produce chiral biomolecules are stereospecific in nature. Once an orientation has been favored, shifting to the opposite is both difficult and unnecessary. Homochirality is thus the simplest and most parsimonious evolutionary case.