Gas-phase anion photoelectron spectroscopy (PES) is combined with global structural searches and electronic structure calculations at the hybrid Becke 3-parameter exchange functional and Lee-Yang-Parr correlation functional (B3LYP) and single-point coupled-cluster with single, double, and perturbative triple excitations (CCSD(T)) levels to probe the structural and electronic properties and chemical bonding of the B4O4 (0/-) clusters. The measured PES spectra of B4O4 (-) exhibit a major band with the adiabatic and vertical detachment energies (ADE and VDE) of 2.64 ± 0.10 and 2.81 ± 0.10 eV, respectively, as well as a weak peak with the ADE and VDE of 1.42 ± 0.08 and 1.48 ± 0.08 eV. The former band proves to correspond to the Y-shaped global minimum of Cs B4O4 (-) ((2)A″), with the calculated ADE/VDE of 2.57/2.84 eV at the CCSD(T) level, whereas the weak band is associated with the second lowest-energy, rhombic isomer of D2h B4O4 (-) ((2)B2g) with the predicted ADE/VDE of 1.43/1.49 eV. Both anion structures are planar, featuring a B atom or a B2O2 core bonded with terminal BO and/or BO2 groups. The same Y-shaped and rhombic structures are also located for the B4O4 neutral cluster, albeit with a reversed energy order. Bonding analyses reveal dual three-center four-electron (3c-4e) π hyperbonds in the Y-shaped B4O4 (0/-) clusters and a four-center four-electron (4c-4e) π bond, that is, the so-called o-bond in the rhombic B4O4 (0/-) clusters. This work is the first experimental study on a molecular system with an o-bond.