I think its probably a combination, although I believe the right brain may be more responsible for spatial organization skills. I was always a left brain type, but I now have what is sometimes called a diffuse brain injury or a mild traumatic brain injury. My verbal skills are good enough but shapes and numbers have become my nemesis. I do the 9 cube puzzles which drive me crazy because you, in effect, have to remember shapes to solve them - the shapes of the numbers. If there was a word in each square, I imagine I would do better. Its really true that if you don't use it you'll lose it. When I got really sick at the end, I didn't have the energy to leave my house. I spent a lot of time just playing scrabble on my computer, but I didn't venture out to new places or sit around staring at geometric shapes.
My neurologist is actually testing a group of his patients to see if soduku puzzles played 3 times a day help patients such as myself show improvement in spatial tasks. Since I scored a whopping one percent in spatial recall on my neuropsych eval, I'm open to anything. It can't hurt, anyhow!