Using measurements obtained by the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite, relationships between layer-integrated depolarization ratio (delta) and layer-integrated attenuated backscatter (gamma) are established for moderately thick clouds of both ice and water. A new and simple form of the delta-gamma relation for spherical particles, developed from Monte Carlo simulations and suitable for both water clouds and spherical aerosol particles, is found to agree well with the observations. A high-backscatter, low-depolarization delta-gamma relationship observed for some ice clouds is shown to result primarily from horizontally oriented plates and implies a preferential lidar ratio - depolarization ratio relation in nature for ice cloud particles containing plates.