Imagine a clock on the wall that can be adjusted to have either the 3 or the 9 pointing straight up, as well as the 12.
My best idea so far is to use a disc of plywood with through holes at 9-12-3 and a suitable lag bolt to hold both the rotating plywood disc and the item of interest also mounted to the plywood disc. One fixed hole in the wall at 12, and then a bit of dowel to keep the item in one of three possible alignments.
Thanks