Hi flippyfloop attached here is another version with the seam rotated like Burr's.

My method is Edit > Separate on the sphere face. Do an "untrim" by selecting one edge of it, then Ctrl+A for select all then Delete. Then Transform > Rotate > Rotate axis to rotate the sphere, then Edit > Trim and use the "Select naked edges" shortcut key (http://moi3d.com/forum/index.php?webtag=MOI&msg=6051.2) in the step where you are selecting cutting objects. Then Edit > Join to glue it back together.

- Michael