In BnNn cages or tubes, when the quasi-borazine rings are attached to each other through a pair of common atoms of B and N, the bonding structure is named class A. On the other hand, there are some BnNn rings including a covalent bond between two atoms of B and N, which are named class B. In all previous studies, both reports of synthesis and theoretical calculation of boron nitride tubes and cages, the quasi-borazine units are attached together like class A. There are no theoretical or experimental reports from class B compounds except for a brief study in our previous works (Struct. Chem. 2012, 23, 551-580; J. Phys. Chem. C 2010, 114, 15315.). In this study, we have used two kinds of boron nitride rings from a twisted BN sheet in the same chirality created by different mechanisms. For (4, 4) chirality, the molecules B16N16 and B15N15 are found to respectively represent class A and B, and for (5, 5) chirality the molecules B20N20 and B18N18 are respectively again of class A and B. The structure of class A rings is similar to boron nitride tubes, but we have shown that it is impossible to produce a macromolecule of class B form as tubes or cages, because there is much more instability and intermolecular tension in macro forms of class B. This is the main reason that the class B molecules are rare and, because of their small size, have not yet been synthesized, although we have some suggestions for the synthesis of these kinds of molecules. The stability and electromagnetic properties with hybrid density functional theory using the EPR-III and EPR-II basis sets for explanation of hyperfine parameters and spin densities, electrical potential, and isotropic Fermi coupling constant of these rings have been studied by the nonbonded interaction models. Normal mode analyses including aromaticity have been investigated by using the nucleus independent chemical shift values at the ring center. Interaction energy and gain in energy aid in describing the stability that is promoted upon gradual binding with molecular hydrogen, and a linear relationship occurred between them.