Here's a hint that helped me understand much of the mystery of phone cord kinks:
Notice how in bhouston's image above the coil on the left of the kink has opposite spin as the coil on the right. With this kind of kink there is no solution but to completely re-coil one of the sides.
I think you could "fold" the whole cable back on itself at that point. Instead of unwinding around the axis of the helix, unwind around an axis at a right-angle.
No, dual is correct. There's no local way to unkink the phone cable in that image. Just notice that the orientation (whether a cord spirals clockwise or counterclockwise as you travel along it) doesn't change in a properly unkinked cord, but the left and right side of the cord in the photograph have opposite orientations. Whatever manipulation you do locally around the kink (even if it involves rotating the whole rest of the cord around rigidly) won't change the orientation of either side.
A corollary to this is that the kink in the picture wasn't created locally, and is not the kind of kink you accidentally create. Though you can create a stretch of mis-oriented cord by trying to fix what starts as a local kink, but that requires fiddling with that entire section of cord.
