Structural studies of fullerene-like Si(60) and Ge(60) cages using ab initio methods were augmented by density functional tight-binding molecular dynamics (DFTB-MD) simulations of finite temperature effects. Neither the perfect I(h) symmetry nor the distorted T(h) structures are true minima. The energies of both are high relative to distorted, lower symmetry minima, C(i) and T, respectively, which still preserve C(60)-type connectivity. Both Si(60) and Ge(60) favor C(i) symmetry cages in which Si and Ge vertexes exhibit either near-trigonal or pyramidal geometries. These structural variations imply significant reactivity differences between different positions. The small magnetic shielding effects (NICS) indicate that aromaticity is not important in these systems. The inorganic fullerene cages have lower stabilities compared with their carbon analogs. Si(60) is stable towards spontaneous disintegration up to 700 K according to DFTB-MD simulations, and thus has potential for experimental observation. In contrast, Ge(60) preserves its cage structure only up to 200 K.