We demonstrate the self-formation of hexagonal nanotemplates on GaAs (111)B substrates patterned with arrays of inverted tetrahedral pyramids during metal-organic vapor phase epitaxy and its role in producing high-symmetry, site-controlled quantum dots (QDs). By combining atomic force microscopy measurements on progressively thicker GaAs epitaxial layers with kinetic Monte Carlo growth simulations, we demonstrate self-maintained symmetry elevation of the QD formation sites from three-fold to six-fold symmetry. This symmetry elevation stems from adatom fluxes directed towards the high-curvature sites of the template, resulting in the formation of a fully three-dimensional hexagonal template after the deposition of relatively thin GaAs layers. We identified the growth conditions for consistently achieving a hexagonal pyramid bottom, which are useful for producing high-symmetry QDs for efficient generation of entangled photons.