Bowl-shaped pi-conjugated compounds offer the possibility to study curvature-dependent host-guest interactions and chemical reactivity in ideal model systems. For surface-adsorbed pi bowls, however, only conformations with the bowl opening pointing away from the surface have been observed so far. Here we show for sumanene on Ag(111) that both bowl-up and bowl-down conformations can be stabilized. Analysis of the molecular layer as a function of coverage reveals an unprecedented structural phase transition involving a bowl inversion of one-third of the molecules. On the basis of scanning tunneling microscopy (STM) and complementary atomistic simulations, we develop a model that describes the observed phase transition in terms of a subtle interplay between inversion-dependent adsorption energies and intermolecular interactions. In addition, we explore the coexisting bowl-up and -down conformations with respect to host-guest binding of methane. STM reveals a clear energetic preference for methane binding to the concave face of sumanene.