Hi - I posted this question as a reply to the Map Viewer beta help doc board here:
In the earlier reply above, I thought my hosted feature layer collection in Florida West State Plane HARN (2882) coordinate system was not being properly transformed by map viewer beta to overlay correctly.
It turns out that the issue is not with the feature layer collection, but rather with a tiled(cached) image service we use as part of our local image service basemap.
The interesting thing is that while this 3-layer image basemap (street vector tile in WebMercator, tile cache image service in WebMercator, esri world imagery) works in the Beta viewer, the tile cache image service in WebMercator 'shifts' along the x-axis.
I know support is partial for image services in Beta, but for a lark I tried replacing the WebMercator tile cache image service in my webmap with my State Plane HARN tile cache image service (2882) and saved as a new webmap/basemap. No x-axis shift! How odd!
I don't know what the cause is, but when I created my WebMercator image service , I used the mosaic dataset reproject function in Pro so that I would not have to re-project the source tiffs. I suppose somehow Beta is not 'honoring' that function, even though the service is web mercator: