Boron oxide clusters are electron-deficient species with novel structures and bonding, in which the emergence of rhombic and boroxol rings is of interest. We report on computational prediction of the global-minimum structures for two boron oxide clusters: B4O5 and B4O5-. These structures differ distinctly, as established through global machine searches and electronic structure calculations at B3LYP and single-point CCSD(T) levels. While B4O5 neutral cluster has a rhombic B2O2 core, the B4O5- anion features a boroxol B3O3 ring. One electron completely changes the potential landscapes. Bonding analyses show that the 4 pi electron-counting is crucial for a rhombic B-O cluster, in contrast to pi sextet for a boroxol ring, which underlies the competition between rhombic and boroxol rings in B4O5/B4O5- clusters. A possible pathway for rhombic-to-hexagonal transformation is proposed based on intrinsic reaction coordinate calculations. Anion B4O5- cluster, a new member of the inorganic benzene family, is among the smallest B-O species with a free-standing boroxol ring, governed collectively by composition and electron-counting.