The existence and structure of BeCN2, the lightest representative of II-IV-V2 compounds, have for long remained unsolved, although previous theoretical studies have relied on assuming chemical similarity toward the known wurtzite-type BeSiN2. To solve the BeCN2 puzzle, we have now explored its potential-energy surface and here predict two additional polymorphs with space groups Cmc21 (porous phase) and Pmc21 (graphitic phase) in addition to another I4̅m2 type (carbodiimide-like), which is only slightly higher in energy than the wurtzite type. The phase diagram constructed from density-functional theory shows the Cmc21-type to be the ground state, stable in terms of the Gibbs energy under standard conditions, whereas the Pmc21- and I4̅m2-types are high-temperature phases; the wurtzite type, however, is the high-pressure phase. The kinetic barrier between the porous and graphitic phases is small, about 4 kJ mol-1, but larger toward the carbodiimide type, 25 kJ mol-1, and the wurtzite type, 28 kJ mol-1. Chemical-bonding analysis further reveals how beryllium and carbon induce structural diversity. As regards the second-lowest Pmc21-type, a monolayer of such graphitic BeCN2 shows the potential of photoelectrochemical water splitting, while a bilayer configuration should exhibit ferroelectricity with a polarization of 0.75 pC m-1. Further electronic-structure data of the four polymorphs indicate their potential for nonlinear optics.