Duncan I have cleaned my corrector on a couple of occassions (LX200) but only when it's really dirty. The procedure I follow is as follows.
1. Set up on a flat surface with plenty of room and have a soft towel folded ready to accept the corrector plate. Soft clean gloves will save you a lot of extra work and prevent oil transfer from your skin.
2. Point the scope upward for easy access and carefully loosen the bolts on opposite sides a little at a time (1/8 turn) until they are free enough to be wound out by hand.
3. Remove the retainer and make sure your corrector is marked so you can re-align it again, a must on a SCT. (mine has a thick white line with a black line in the centre drawn by a fine marker).
4. You need to be very careful when removing the corrector as it gets stuck on the cork spacers. I use a pop stick to gently apply pressure around the corrector until it comes loose. Before you take it out note where the cork spacers are and make sure they go back in the correct position if they come loose.
5. Get a good grip on the secondary mirror holder and lift the corrector out placing it face down on the towel and rotate the scope so it is pointing down to prevent dust from settling on the primary. I have a small plastic cap which fits perfectly over the secondary baffle which I use to cover and protect the secondary mirror. Think I got it off a new turbocharger when I was involved in racing

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6. Try not to get anything on the secondary mirror as getting it off again will be very difficult without damage. I first wash the excess dust off under the tap and then rinse using de-ionised water then let it dry. I use the bintel UHTC cleaning fluid and cotton rags for the final clean on the corrector. The only trick here is to wipe in straight lines and use a new surface each time changing rags regularly.
7.When you are satisfied that it is totally clean, flip the scope back up and replace the corrector making sure it is aligned properly with the marks.
8. Replace the retainer and all the bolts by screwing them in with your fingers until you just feel light resistence. Use an allen key to tighten each bolt 1/8th turn at a time. It is important that you do opposites as this keeps the pressure even and does not stress the glass which will crack. You only need to tighten until you feel light resistence on the allen key, very little pressure is needed and the glass will stick to the cork. Use your noggin here as a busted corrector means a useless scope. Thats it.
I have never found reason to clean either mirror as they don't seem to attract dust like the corrector. I would never physically touch a mirror and would only ever rinse them with water if they got really really dirty. If you remove the secondary holder make sure you put it back in the same alignment as I am led to believe this is also essential.
Good Luck
Mark