Many quarkoniumlike states have been observed above open-flavor thresholds, but their organization and internal structure remain unsettled. We study the isoscalar hidden-charm and hidden-bottom sectors in Born-Oppenheimer effective field theory (BOEFT), between the spin-isospin averaged S + S and S + P thresholds. At leading order, heavy-quark spin decouples, and the quarkonium static potential mixes through string breaking with the lowest tetraquark/open-flavor BO potentials of the same quantum numbers. These potentials are constrained by QCD symmetries, their short- and long-distance behavior, and lattice-QCD data. The only calibrated parameter is the lowest 1^{–} adjoint meson mass, fixed from the shallow multiplet associated with the _{c1}(3872). Using T-matrix, K-matrix, and complex-scaling methods, we determine bound states and resonance poles, their masses, pole widths from the included nonstrange S + S channels, normalized pole couplings, and prescription-dependent quarkonium-open-flavor composition measures. Uncoupled hybrid BOEFT multiplets are included as reference levels. The spectrum exhibits a common heavy-quark-spin-symmetry multiplet organization. Most poles are predominantly quarkonium resonances localized at relatively short distances, with the largest open-flavor components closest to threshold. The same equations also generate shallow, spatially extended, open-flavor-dominated states with molecular long-distance characteristics. Their binding energies, radii, and small quarkonium components are highly sensitive to the adjoint meson mass, whereas the higher spectrum is more stable. In addition, the equations predict a more compact open-flavor-dominated resonance in the l = 0 bottomoniumlike sector, approximately 50 MeV above the common spin–isospin averaged threshold, with a $P$-wave open-flavor component and a 0^{-+}, $^{–} BOEFT multiplet structure. Together with the hybrid reference levels, the spectrum provides multiplet assignments for most experimental candidates. States not naturally accommodated point to the need for hidden-strange and S + P tetraquark/open-flavor BO sectors and for hybrid-tetraquark and hybrid-quarkonium mixings.