The mechanism of inversion of lead phthalocyanine (PbPc), possessing a shuttlecock shape, was studied in detail within the bonding evolution theory framework. We have found that reaction pathway calculated using hybrid density functional theory (DFT) method, B3LYP, consists of five structural stability domains, which are connected by four bifurcation points. Reorganization of Pb's basins with pronounced role of core ones is the basis of the catastrophes identified, whereas basins belonging to other atoms are almost not involved in the electronic structure changing. These results provide the new topological picture of processes underlying the conformational transitions of shuttlecock-shaped metal phthalocyanines adsorbed on surfaces.