What are the hardest types of puzzles involving spatial manipulation?

Chess. It involves the conquest of space, and time. It requires that short term tactics line up with, and are in service of long term strategies. Great moves, must be made with great timing.

If you do a bit of research on mechanical puzzles, I'm sure you'll find something that will interest you. There are extremely high level 'burr' puzzles, such as the Millenium puzzle, taking 1000 moves to take the first two pieces out. http://puzzlewillbeplayed.com/13BoardBurr/Millenium/ A burr puzzle is a (usually) cubic figure constructed by smaller shapes that interlock. Some, such as this puzzle, can take multiple moves before any piece is freed from the structure. Sometimes rotations and other nonlinear moves are required (shameless plug; you can see some of my burr designs here: http://puzzlewillbeplayed.com/-/designer/Hu.xml ) There are disentanglements, variations of which I'm sure you've come across before; wire puzzles, rope or knot puzzles, etc. While the mass-produced designs are usually not very difficult, it is possible to find some hard puzzles of to type. 'Twisty' puzzles are extremely difficult. A simple example of a twisty puzzle is the Rubiks cube, where the aim is to re-sort the colored faces. Complex twisty puzzles can be non-cubic, irregular, and can 'shape shift', where it is difficult even to restore the puzzle to its original shape. There are lots of other types of mechanical puzzles, but the ones I've highlighted above are the ones that tend to lend themselves to more difficult designs.