This fascinating puzzle is called Orb in the USA and Orb-It in the UK. The puzzle consists of a ball with two small rings of beads near the poles, and two larger rings parallel to and near the equator. The four rings have different colours. The bead rings rotate around the ball. The ball itself is divided vertically into two halves. By rotating one half the rings of beads become either one long spiral, or two loops (looking a bit like a tennis ball). In the solution below the spiral position is never used.
The beads in the polar rings are blue or yellow. The remaining beads are red or green.
It might be interesting to quote from the afterword of the Rubik's Cubic Compendium [p213] written by David Singmaster.
A further type of closely related puzzle is what I call the 'switchable cycle' puzzles. Here one has several disjoint cycles of pieces with some mechanism for splitting the cycles. The best known example was the Orb-It (UK) or Orb (US), which had four parallel tracks on a sphere. One hemisphere could be turned to eight positions with respect to the other, causing the half-tracks to join up in several different ways. This was marketed in both Europe and the United States.
It was invented by Christopher C. Wiggs & Christopher J. Taylor, and they filed for a US patent on 16 February 1982, which was granted in 19 November 1985, US 4,553,754. An earlier British patent, GB 8113543, dates from 1 May 1981.
There are 8 beads in each of the polar rings and 20 in the equatorial ones, which gives: 56! / (8!220!2) = 73,888,773,475,012,113,089,523,051,000 or 7.4·1028 positions. This considers rotations of the rings as the different positions, and does not include the other loop shapes.
Phase 1: Separate the polar/equatorial beads. In this phase the blue/yellow beads are placed in the polar rings, and the red/green beads are therefore displaced to the equatorial rings.
Note that the above method doesn't just work with a single bead in each ring. It works just as well with a pair of adjacent beads in each ring, or even three of four.
Phase 2: Separate red and green, and similarly blue and yellow.
Note that the above moves don't just work with a single bead in each ring. It works just as well with a pair of adjacent beads in each ring, or even three or more.