The structural characteristics and properties of three new carbon phases (LA9, LA10, CA12), which have a diamond-like structure and atoms located in crystallographically equivalent positions, are described. The model mechanism of LA9 and LA10 formation is the linking of L-6 and L4-8 graphene layers, respectively, and phase CA12 can be formed by linking C-4 tetrahedral clusters. Phases LA9, LA10, and CA12 can also be formed as a result of the polymorphic transformations of three-dimensional graphite phases, when all atoms transform from a three-coordinated into a four-coordinated state. LDA-DFT calculations of the LA9, LA10, and CA12 phases are used to find their geometrically optimized structures and properties (density, total energy, density of states). In addition, powder X-ray diffraction patterns are calculated for these phases and possible methods of their synthesis are analyzed.