This review aims to summarize fundamental concepts and principles of isolobal and isospin analogy between organic and inorganic molecules with local spins. The isolobal analogy based on the extended Hückel (EH) molecular orbital (MO) model has been successfully applied for organic and organometallic compounds with closed-shell eight and eighteen electron rules. The EHMO with spatial symmetry has been employed for confirmation and elucidation of the orbital symmetry conservation rules for concerted reactions. On the other hand, the isospin analogy based on the Heisenberg spin Hamiltonian (HSH) model has been proposed for conceptual bridges between organic and inorganic open-shell molecules and clusters with the magnetic symmetry given by the spin rotation and time-reversal (spin inversion) symmetries. The spin-correlation diagrams by the HSH model are available for elucidation of exchange-allowed and -forbidden radical reactions. The broken-symmetry (BS) MO and density functional theory (DFT) models are constructed with spatial, spin rotation and time-reversal symmetries for open-shell species with one-dimensional (1D), 2D and 3D spin orbital structures. Both isolobal and isospin analogy based on the BS MO and DFT models have been applied for elucidation of electronic mechanisms of oxygenation reactions by active oxygen, oxyradicals and high-valent transition-metal oxo species with the oxyl-radical character. The isolobal and isospin analogy are also applicable for examination and investigation of possible mechanisms of the oxygen-oxygen (OO) bond formations for water oxidations catalyzed with the native CaMn4Ox cluster and artificial 3d transition-metal complexes. To this end, BS DFT methods have been applied to elucidate geometrical, electronic and spin structures of the CaMn4Ox (X = 5, 6) clusters in the Si (i = 1 ~ 4) states of the Kok cycle for water oxidation on the basis of the geometrical structures by the X-ray diffraction (XRD and SFX XFEL) methods. The computational results have been compared with spectroscopic (EXAFS, XES, EPR) results for elucidation of scope and reliability of the BS hybrid DFT followed by DLPNO-CCSD(T) level of theory for strongly correlated electron systems (SCES) such as the Mn and Fe-oxide complexes. Interplay between theory and experiment is effective and powerful for unraveling secrets of the water oxidation in OEC of PSII and related artificial systems. Fundamental concepts revealed by the interplay are applied to design of bio-inspired artificial Z-schemes for conversion of solar energy to chemical energy.