Shuttlecock shaped metal-phthalocyanine (MPc) can adsorb on a substrate surface having the central metal atom either down or up and the possibility of reversible switching between these two adsorption configurations shows great promise for use in nanomechanical devices. Using density functional theory we investigate the mechanism of the internal conformational inversion of germanium, tin and lead phthalocyanine in terms of the geometry, energy barrier of the reaction, and redox properties of the central metal atom. We have found the same mechanism of inversion for GePc and SnPc but a different one for PbPc. Inversion proceeds through two transition states, separated by a planar local minimum, for GePc and SnPc, but through one transition state distorting the phthalocyanine macrocycle for PbPc. The energy barrier of inversion is 4.27 eV for PbPc and 2.12 and 3.16 eV for GePc and SnPc, respectively. Such high barriers are unlikely to be overcome at normal experimental conditions, and in many cases alternative explanations for switching between "up" and "down" conformation need to be sought, such as ionization assisted inversion or even flipping over of the molecules. Our calculations show that the inversion of GePc and SnPc is accompanied by reversible two electron oxidation (M(II)<--> M(IV)) of the metal atom, through intersystem crossing. The difference in mechanism of inversion for GePc (SnPc) and PbPc is assigned to the different nature of the central metal atom.