We have developed a modular, efficient catalytic arene-norbornene annulation (CANAL) chemistry to construct hydrocarbon ladder molecules containing four-membered rings using bromoarenes and (oxa)norbornenes as building blocks. CANAL chemistry enabled a new family of rigid yet soluble hydrocarbon ladder polymers consisting of geometrically variable backbone contortion. These ladder polymers exhibit remarkably high thermal stability and surface area, and have been explored as membrane materials for chemical separations. Tuning the hydrocarbon backbone structures resulted in unprecedented combination of high permeability and high selectivity for many industrially important gas separations. CANAL chemistry also enabled the access to a range PCHs with fused cyclobutadienoids (CBDs) of variable degree of antiaromaticity. The influence of CBDs on the local antiaromaticity and aromaticity, bonding structures, and redox properties has been studied both experimentally and computationally. These new CBD-containing PCHs also show distinct reactivity from the classic [N]phenylenes and can be used to synthesize various curved carbon structures with defined topologies.