Orbits
Each of the twenty four center pieces falls in to one of four orbits. Each orbit contains one center piece of each colour. This is always true, even when the cube is scrambled up.
To visualize an orbit, find any center piece and you should see there are only two twists which move it. One twist will bring it around its orbit in one direction, and the other will take it the other way. If you follow the path set out by the twists you should see that there are only six places that center can fall in, one on each face.
Jumbling
The puzzle jumbles when you turn one edge through one third of a turn and then twist an intersecting edge. Most jumbles end with the cube off-shape, but if it is started with two edges which touch on a point, then a symmetric jumble appears and you can put that back in the wrong way round which is what I'm going to call jumbling.
Jumbling breaks the orbit rule and allows centers to shift inbetween orbits. If the orbits are not properly distributed then the cube is unsolvable because one of the orbits will be missing a colour.
Interestingly the corners are never put in to an unsolvable state by jumbling.
What a jumble does
A jumble represented by this strange icon:
has the following effect.
The magenta marked centre is swapped with the centre on the bottom face directly beneath it (shown in shadow). The two cyan edges are swapped with each other.
Note that the two cyans are on the same orbit. The magentas are on different orbits from each other and from the cyans.
Sorting out the orbits
Sorting out the orbits can be done quite quickly by checking that five of the six centers can be constructed validly. You can also fix the orbits at the same time as you check, so that by the time you have checked each one you can jump straight in to step two of the solution.
You have to pick each colour in turn. The order doesn't matter. Let's say you start with blue. Try to complete step one of the solution and build up the blue center. If you can place all four blue pieces together to form a diamond then you know all the blues are in separate orbits and you can continue on to a different colour. If you find that two blues both occupy the same place on a center, they are in the same orbit and shouldn't be.
If you have two matching colours in the same orbit, then that orbit must also be missing at least one colour. Look around the orbit and find a colour that is missing from it. Let's say your orbit is blue - blue - red - green - yellow - white. That means you need to swap one of the blues for an orange, because orange is missing from this orbit. If you swap it with the first orange you find, there's a good chance you'll take an orange from an orbit where it is unqiue. Instead, find an orange that has another orange on its orbit, an orange with a duplicate. You can do this by just looking carefully or by trying to construct an orange center somewhere else on the cube. Once you find an orange duplicate, you need to swap a duplicate blue and a duplicate orange.
To swap them place them on the magenta positions shown on the jumble above. Don't worry about the cyan positions because they are both on the same orbit so they won't reverse any progress you've made on correcting your orbits. It may take a number of twists to find out where you can jumble them correctly but don't worry, you can always arrange two pieces on separate orbits in this position somewhere on the cube. Once you jumble them, go back to making your blue center until all four blue center pieces are on different orbits. Watch out: don't just assume you can build a center if you swap out your last duplicate.
If you have trouble placing them opposite, here are some example twists which place the pieces in the correct opposite position for jumbling. You'll have to keep your fingers on them because the orientation is different for each of them.
Once you have done that for five out of the six centers, you know your orbits are all unique. You don't need to do the sixth colour because once you have confirmed five of the colours as being in correct orbits the sixth one has to be correct. You can also check the orbits by reading the colours in three of the orbits and checking each colour is unique for that orbit. Again if three of the orbits all contain unique colours, the last orbit has to contain unique colours too.
Parity
Once you have one of each colour in each orbit, you can solve the cube. However there is a chance you will get a parity situation as shown.
This is a parity problem where the orbits are good but have bad parity. It can be fixed with this algorithm, which involves a jumble half way through.
That will solve the cube entirely. If you get the mirror image case, or any other parity case, this algorithm will still make the cube solvable.