Tetragonal and hexagonal hybrid sp$^3$/sp$^2$ carbon allotropes C5 were proposed based on crystal chemistry and subsequently used as template structures to identify new binary phases of the B–N system, specifically tetragonal and hexagonal boron nitrides, B$_2$N$_3$ and B$_3$N$_3$. The ground structures and energy-dependent quantities of the new phases were computed within the framework of quantum density functional theory (DFT). All four new boron nitrides were found to be cohesive and mechanically (elastic constants) stable. Vickers hardness (HV), evaluated by various models, qualified all new phases as superhard (Hv > 40 GPa). Dynamically, all new boron nitrides were found to be stable from positive phonon frequencies. The electronic band structures revealed mainly conductive behavior due to the presence of π electrons of sp$^2$-like hybrid atoms.