Prevent rotated UV islands when unwrapping a quad sphere

Question

I've created a quad sphere using several different methods (sphere -> subdiv surface; sphere -> subdiv surface -> cast). I then marked 12 seams and tried unwrapping it using several methods. The best result for my purposes I get using the regular "unwrap" method. However, the generated UV islands are slightly rotated. Is there a way to prevent that? I Googled a bit and found that I could "pin" certain nodes. But what I tried, I couldn't get nice UV islands. Below I've added some screenshots of my situation. Does anybody know how I could get a result matching the last screenshot?

Here's my result. As you can see the islands are slightly rotated:

Here are the seams I have defined:

And here's the result I'd like to get. This UV I got from an already properly unwrapped triangle-based model I got from the internet:

Answer 1

I actually solved it myself. I installed the Add Mesh: Extra Objects extension that comes with Blender. Then I created a new quad sphere by doing Add -> Round Cube. I expanded the properties dialog that appears in the lower left of Blender and selected Quadsphere from the Operator Presets dropdown. When you mark the proper seams (see my opening post) and unwrap it, the UV islands appear straight. I don't know why this differs from my own method, but at least it's working.