  Computer simulations give precious insight into the microscopic behavior of
supercooled liquids and glasses, but their typical time scales are orders of
magnitude shorter than the experimentally relevant ones. We recently closed
this gap for a class of models of size polydisperse fluids, which we
successfully equilibrate beyond laboratory time scales by means of the swap
Monte Carlo algorithm. In this contribution, we study the interplay between
compositional and geometric local orders in a model of polydisperse hard
spheres equilibrated with this algorithm. Local compositional order has a weak
state dependence, while local geometric order associated to icosahedral
arrangements grows more markedly but only at very high density. We quantify the
correlation lengths and the degree of sphericity associated to icosahedral
structures and compare these results to those for the Wahnstr\"om Lennard-Jones
mixture. Finally, we analyze the structure of very dense samples that partially
crystallized following a pattern incompatible with conventional fractionation
scenarios. The crystal structure has the symmetry of aluminum diboride and
involves a subset of small and large particles with size ratio approximately
equal to 0.5.
