Arbitrary angles or slices for split screens
I don't know of any software that makes this trivial to do, however After Effects can certainly accomplish it.  You would need to make multiple masks that would show portions of the same video and then position them however you want.  You could even derive the masks from a shape imported from Adobe Illustrator that you could slice up however you wanted to ensure that you get full coverage.

This technique is still non-trivial and would take some time to do manually, but you could then manipulate each of the shape layers however you wanted using motion keyframes in After Effects.