Six La ABB'O6 ( 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 P21 /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 d8-d3 or d7-d3, not one of the six compounds shows ferromagnetism as predicted by the Goodenough-Kanamori rules. LaSrNiOsO6, LaSrNiRuO6, and LaCaNiRuO6 display long-range antiferromagnetic order, while LaCaNiOsO6, LaCaCoOsO6, and LaSrCoOsO6 exhibit spin-glass behavior. These compounds are compared to the previously studied LaCaCoRuO6 and LaSrCoRuO6, 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 t2g orbitals of the 3d ion.