Catena(dimethylammonium-bis(mu(2)-chloro)-chlorocuprate), (CH3)(2)NH2CuCl3, forms chains of Cu2Cl62- bifold dimers linked along the structural chain axis by terminal chlorides forming long semicoordinate bonds to adjacent dimers. The structural chains are separated by dimethylammonium ions that hydrogen bond to chloride ions of the dimers. A structural phase transition below room temperature removes disorder in the hydrogen bonding, leaving adjacent dimers along the chain structurally and magnetically inequivalent, with alternating ferromagnetic and antiferromagnetic pairs. The coupled dimers are magnetically isolated from each other along the structural chain axis by the long semicoordinate Cu-Cl bond. However, the dimers couple to like counterparts on adjacent chains via nonbonding Cl center dot center dot center dot Cl contacts. The result is two independent magnetic chains, one an alternating antiferromagnetic chain and the other an antiferromagnetic chain of ferromagnetically coupled copper dimers, which run perpendicular to the structural chains. This magnetostructural analysis is used to fit unusual low-temperature (1.6 K) magnetization vs field data that display a two-step saturation. The structural phase transition is identified with neutron scattering and capacitance measurements, and the X-ray crystal structures are determined at room temperature and 84 K. The results appear to resolve long-standing confusion about the origins of the magnetic behavior of this compound and provide a compelling example of the importance of two-halide magnetic exchange.