Understanding the conditions that favour crystallization or vitrification of liquids has been a long-standing scientific problem(1-3). Another connected, and not yet well understood question is the relationship between the glassy and the various possible crystalline forms a system may adopt(4,5). In this context, B2O3 represents a puzzling case. It is one of the best glass-forming systems despite an apparent lack of low-pressure polymorphism. Furthermore, the system vitrifies in a glassy form abnormally different from the only known crystalline phase at ambient pressure(6). Last but not least, it never crystallizes from the melt unless pressure is applied, an intriguing behaviour known as the crystallization anomaly(7-9). Here, by means of ab initio calculations, we discover the existence of previously unknown B2O3 crystalline polymorphs with structural properties similar to the glass and formation energies comparable to the known ambient crystal. The energy degeneracy of the crystals, which is high at ambient pressure and suppressed under pressure, provides a framework to understand the system's ability to vitrify and the origin of the crystallization anomaly. This work reconciles the behaviour of B2O3 with that from other glassy systems and reaffirms the role played by polymorphism in a system's ability to vitrify(10,11). Some of the predicted crystals are cage-like materials entirely made of three-fold rings, opening new perspectives for the synthesis of boron-based nanoporous materials.