Is it possible to re-use a Cut Region definition so multiple turning operations can see this same region without having to redefine it over and over?
Yes, this can be done, but the Cut Region must be set up as a geometry parent beforehand. Under the turning workpiece, define a CONTAINMENT object using Insert; Geometry; "turning" template; CONTAINMENT subtype.
In here, define the Cut Region as usual. Then put operations under this containment parent so they will all inherit that same region definition. The final structure of the Geometry view of the Operation Navigator would look something like this: