We employ ab-initio electronic structure calculations to study 60 LiMgPdSn-type (also known as LiMgPdSb-type) quaternary Heusler compounds. All compounds obey the Slater-Pauling rule with diverse electronic and magnetic properties. 41 compounds are found to be half-metals, 8 spin-gapless semiconductors, and 9 semiconductors. CoVTiAl and CrVTiAl compounds are identified as ferromagnetic and antiferromagnetic semiconductors, respectively, with large energy gaps in both spin directions. All magnetic compounds are expected to have high Curie temperatures making them suitable for spintronics/magnetoelectronics applications.