Six La ABB'O ( A = Ca, Sr; B = Co, Ni; B' = Ru, Os) double perovskites were synthesized, several for the first time, and their crystal structures and magnetic behavior were characterized with neutron powder diffraction and direct-current and alternating-current magnetometry. All six compounds crystallize with P2 /n space group symmetry, resulting from a a c octahedral tilting and complete rock salt ordering of transition-metal ions. Despite the electronic configurations of the transition-metal ions, either d-d or d-d, not one of the six compounds shows ferromagnetism as predicted by the Goodenough-Kanamori rules. LaSrNiOsO, LaSrNiRuO, and LaCaNiRuO display long-range antiferromagnetic order, while LaCaNiOsO, LaCaCoOsO, and LaSrCoOsO exhibit spin-glass behavior. These compounds are compared to the previously studied LaCaCoRuO and LaSrCoRuO, both of which order antiferromagnetically. The observed variations in magnetic properties can be attributed largely to the response of competing superexchange pathways due to changes in B-O- B' bond angles, differences in the radial extent of the 4d ( B' = Ru) and 5d ( B' = Os) orbitals, and filling of the t orbitals of the 3d ion.