Photometric and spectroscopic analyses have shown that the Galactic bulge cluster Terzan 5 hosts several populations with different metallicities and ages that manifest as a double red horizontal branch (HB). A recent investigation of the massive bulge cluster NGC 6569 revealed a similar, though less extended, HB luminosity split, but little is known about the cluster's detailed chemical composition. Therefore, we have used high- resolution spectra from the Magellan-M2FS and VLT-FLAMES spectrographs to investigate the chemical compositions and radial velocity distributions of red giant branch and HB stars in NGC 6569. We found the cluster to have a mean heliocentric radial velocity of -48.8 km s(-1) (sigma = 5.3 km s(-1); 148 stars) and <[Fe/H]> = -0.87 dex (19 stars), but the cluster's 0.05 dex [Fe/H] dispersion precludes a significant metallicity spread. NGC 6569 exhibits light- and heavy-element distributions that are common among old bulge/inner Galaxy globular clusters, including clear (anti) correlations between [O/Fe], [Na/Fe], and [Al/Fe]. The light-element data suggest that NGC 6569 may be composed of at least two distinct populations, and the cluster's low <[La/Eu]> = -0.11 dex indicates significant pollution with r- process material. We confirm that both HBs contain cluster members, but metallicity and lightelement variations are largely ruled out as sources for the luminosity difference. However, He mass fraction differences as small as Delta Y similar to 0.02 cannot be ruled out and may be sufficient to reproduce the double HB.