I've created a map in ArcGIS Online using my own tiled layers (including a PlanetScope image with a rectangular clip) in UTM Zone 10N as a basemap. This worked, however the map is rotated about 15 degrees to the east. I cannot see where to set any rotation, so I don't see how to fix it. Some of the rotation might be explained by having specified the rectangular clip in latitude and longitude and so the UTM will be slightly off from that at different parts of the zone, but by nothing approximating 15 degrees.
Update: I may have the solution, but it's still puzzling. An icon that looked like a clock (see below) showed on the map, and that turned out to be the place to undo the rotation. Clicking it lets you "reset the map orientation" and then clicking the "home" button returns to the default map view, which for some reason is rotated.
If your map was rotated when it was saved, it will retain the rotation next time the map is loaded.