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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.

Threads outer

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

Threads inner

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

Scaled threads -1

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Just to illustrate @batFINGER's comment.
You can flip the profile by scaling it by -1.

  1. Select the profile
  2. Set pivot point to one of the vertices through Pivot point > Active element
  3. Scale it along X axis by -1

enter image description here

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