We report polarized and unpolarized neutron scattering measurements of the magnetic order in single crystals of Na0.5CoO2. Our data indicate that below TN=88 K the spins form a novel antiferromagnetic pattern within the CoO2 planes, consisting of alternating rows of ordered and nonordered Co ions. The domains of magnetic order appear to be closely coupled to the domains of Na ion order, consistent with such a twofold symmetric spin arrangement. Magnetoresistance and anisotropic susceptibility measurements further support this model for the electronic ground state.