Solution:

Phase 1: Make all pieces the same colour.

1. If you can, do any move that makes more pieces of the desired colour.
2. Repeat step a as often as possible, until only most pieces are of the right colour.
3. For each remaining piece that needs to be flipped, put it in pole position and then use 2/-2/2/-2 which flips it.

Phase 2: Parity step
If the pieces are now in an odd permutation, then you will have to flip all the pieces. It is not easy to see whether it is an odd permutation, so you could just try to solve it as in the following phase, and if you are left with just two pieces that need to be swapped then you will have to come back to this phase. Alternatively you could cheat by marking the sides of the ball with the colour of the pieces before getting the puzzle mixed up.
If in an odd permutation then do /6/4/1/1/-2 to flip all the pieces. This sequence preserves the order of the 3 pieces on the small cap. If the piece order does not matter at all, then you can do /5/5/ instead.

Phase 3: Ordering the pieces.
The sequence 1/-1/6/-6/-6/-6/-1 cycles only pieces in positions A, B and C around: The piece in pole position moves to position B, the piece in position B moves to position C, and the piece in position C moves into pole position.

Theoretically you can simply use only this single sequence to solve the ball, but you may want to speed things up by looking up sequences in the table below. To solve the ball with this table, do the following steps:

1. Rotate the ring to bring piece 1 in pole position, and then do -1.
2. Find the piece 2, and note its current position.
3. Look up the sequence in the table that moves a piece from that position to pole position, and do it.
4. Do move -1.
5. Repeat steps b-d for each of the pieces 3-12 to solve it. Make sure that the sequence you choose will not disturb the pieces that are already solved. Sometimes there are several alternative sequences in the table, so that you may be able to solve two pieces simultaneously.

For example, if you have solved pieces 1, 2, and 3 then place them in positions K-M, so that piece 4 will have to be placed in pole position (A) to extend the solved block. Find pieces 4 and 5, and let's suppose they are at positions D and C respectively. Look through the table to find a sequence with D in the second column, and C in the third.

2	D	C	-3/4/1/-2/4/2/

Unfortunately the only such sequence has 2 in the first column, meaning it will disturb the three solved pieces you already have. You will therefore have to choose another sequence, and be content with only solving piece 4 with it.

Of the sequences with at least 3 in the first column, and D in the second, this is the shortest, and therefore the best one to use.

If you end up with a position where you would have to swap the final two pieces, then you have to flip all the pieces round as explained in phase 2.