Is it possible to drag an existing parameter, which is an ordered menu, into Type Properties editor just like we can drag other types of parameters? It seems like this is one of those parms that need to be recreated from scratch.

The second after starting this thread I noticed that there's "Export Parameter to Type Properties" command dug in at the very bottom of the RMB context menu, which will do what I need if Type Properties window is open. Interestingly, its ALT+MMB shortcut does not work for ordered menus.