I have a mesh with bottle threads on the outside. Now I would like to make / invert the threads so they appear on the inside to create the cap. I tried using the boolean modifier and difference with another shape but it doesn't create the threads on the inside (it creates the difference of the threads, a gap). How can I do this with Blender? I'm also willing to do this mathematically if needed.

What I have.

I would like the threads to look like the image below.

If I scale it by -1 the threads aren't created on the inside. See image below.