Cat loves this position, we human, too. Because this is a gesture of relax, day dreaming, doing nothing, to take a break...
To me, a plus of thinking, reviewing, re-considerate of something stuck into the bottleneck. It takes 5 to 30 minutes, or more; ending with a solution or simply nothing, but it's a good chance to see things from upside down, from another angle of view, a real interesting experience.
Especially when you are lying on the beach, looking at the sky above, sandals of people, sand and tree root around, searching for new attractions that are not normally found; your viewing angle is expanding to a great distance, not only the sea in front of you.
Similarly, you will care more other than the tiny problem in front of you, but much more......a person? a long forgotten matter? things happening in other world that far apart? a special moment of something?.......a way out!
If it is so magical, why we don't do it often? Because it takes time - something we are "lack" of; and this magic will never work to the one who refuse to accept the true, even they know the answer is there upside down.