Herein, we introduce a p-type redox conjugated covalent organic polymer (p-PNZ) as a universal and high-rate cathode for diverse dual-ion batteries. By constructing an n-type redox counterpart (n-PNZ) with an analogous reticular structure and redox-site composition, we also attain a comparative platform to probe how the redox-site nature and counterion chemistry affect the rate performance of polymer cathodes. It is disclosed that the charge dilution in p-type redox sites and bulky anions engenders their weak interaction and rapid anion diffusion in electrodes, while the trivial interaction of the solvent with anions facilitates anion desolvation and interfacial charge transfer. Thus, p-PNZ possesses rapid surface-controlled redox kinetics with a high anion diffusion coefficient regardless of its inferior porosity and conductivity relative to n-PNZ. Along with a long cycle life of over 50000 cycles, the p-PNZ-engaged Zn-based dual-ion battery with a dilute electrolyte delivers nearly constant capacities of ∼149 mAh g-1 at various rates of ≤10 A g-1─such an unusual rate capability has rarely been observed previously─and retains ∼99 mAh g-1 at 40 A g-1, surpassing the n-PNZ counterpart and most existing p-type organic cathodes. The p-PNZ cathode can also be applied to build high-rate Li-based batteries, signifying its universality, while the "ready-to-charge" character of p-PNZ enables anode-free dual-ion batteries with a high-rate capability and long lifespan.