Tetrahedral homonuclear tetrameric species are neutral or ionic tetrahedra of chemical elements, true molecular tetrahedra in strict geometrical sense. The aim of this work was to find all true elemental tetrahedra experimentally determined as stable and capable of being transferred from one medium/condition to another, to rationalize their structures and properties, and seek their technological significance. A database of elemental tetrahedra was built based on reliable literature, and a dataset of molecular descriptors was generated. Correlation analysis, hierarchical cluster analysis, principal component analysis, and some regression analysis were carried out. Twelve chemical elements from Groups 13 – 15 have tetrahedral species: anions E48– (Al, Ga, In, Tl), anions E44– (Si, Ge, Sn, Pb), neutral molecules E4 (P, As, Sb), and cations E4+ (P, As, Sb, Bi). These species appear in 39 forms, such as gas phase, solid Zintl phases, thin films, Zintl phase ammoniates, and cage compounds, among others. Zintl phases and forms of P4 and As4 are important for synthesis of new materials, chemical reactions, and other applications. Data compression has shown that the true tetrahedral species are a three-dimensional phenomenon, and that true tetrahedra can be distinguished from fictive tetrahedra. Grouping of the tetrahedra according to their periodic groups, charges, frequency, and diversity class values, tetrahedral size, and metallic character, were observed. Empirical rules for species size and principal components were established for true tetrahedra. Most descriptors are parabolic functions of the class variable. True tetrahedral species partly remain a puzzle and should be further investigated.