Extensive investigation of the newly discovered fullerene-like nested polyhedra (NP) and nanotubules of metal dichalcogenides by scanning tunneling microscopy (STM) and high-resolution transmission electron microscopy is reported. Long-term spontaneous (room temperature) and electron beam assisted crystallization of amorphous precursor into NP or MX(2) (M = W, Mo; X = S, Se) was observed. This, along with other findings, suggests that the NP constitute a new metastable phase between the starting amorphous material and the thermodynamically stable bulk 2H allotrope. Characterization of the apex angles by STM suggests that the apexes of the NP may contain geometrical elements such as triangles and rhombuses, which do not exist in nested carbon fullerenes. The energy bandgap of the NP, determined using STM, was found to be somewhat smaller than that of the bulk 2H phase, and possible explanations for that phenomenon are forwarded.