Sulfur conversion catalysis is an effective strategy to tackle the inherent polysulfide shutting and fast capacity decay in lithium-sulfur batteries. Previous studies identified cobalt phthalocyanine as a unique molecular catalyst with Co-N4 active sites to promote sulfur reduction reactions. Herein, enhancing the Li bonds with the pyridinic N atoms on the phthalocyanine ligand is demonstrated to endow further catalytic activity in sulfur oxidation reactions. This is enabled by peripheral substitution with electron-donating methoxy groups that strengthens the electronegativity of the pyridinic N atoms. DFT calculations and metadynamics simulations identify the key role of Li bonds in reducing the energy barrier of Li2S dissociation. The highly lithiophilic methoxy groups also stabilize the dissociated Li cations in their solvation structure. The dissociated S anion can then be oxidized to higher-order polysulfides or radicals by electron donation to the Co-N4 center, which activates Li2S for subsequent oxidation. Meanwhile, the catalytic activity in sulfur reduction reactions is also improved by a more efficient electron transfer to adsorbed Li2S4, during which the strengthened Li bonds are beneficial in mediating the breakage of the bridging S-S bond. Significant performance improvements are achieved with the bidirectional catalyst under high areal sulfur loading and in Li-S pouch cells.