  The Weyl group $W(\mathrm{E}_6)$ has an odd presentation due to Christopher
Simons as factor group of the Coxeter group on the Petersen graph by deflation
of the free hexagons. The goal of this paper is to give a geometric meaning for
this presentation, coming from the action of $W(\mathrm{E}_6)$ on the moduli
space of marked maximally real cubic surfaces and its natural tessellation as
seen through the period map of Allcock, Carlson and Toledo.
