A recent trip to Moab to photograph this location produced this photo. Known as the False Kiva, it is a protected archaeology site on public land.
It seems to be an unfinished kiva style dwelling. But after my visit there, I would suggest it is a ceremonial site used by ancient tribes.
It is built high above the valley floor, yet several hundred feet below the plateau above in a shallow cave alcove. Undoubtedly chosen by it's builders for it's limited access and astonishing view of the valley below. The trail into the site is not for persons afraid of heights or steep narrow trails just feet from what would be a long fall off the nearby cliff.