If they're layers you've already added to the map, no, at least not without custom scripting. But if they're layers added to the basemap's webmap item, then yes!
Your basemaps are all just web maps shared to whatever group populates the basemap gallery in your AGOL org. You can add any webmap with any number of layers to that same group. So for instance, I might have a web map I've shared to the group named "Cadastral Basemap" for my county. Turning that basemap on could add parcels, zoning, boundaries, etc. Once I switch to another basemap, those layers are then removed.
- Josh Carlson
Kendall County GIS
