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I am surprised that there is no thread for this puzzle so far (or have I missed it?).

The initial grid fill is somewhat easier than it would first appear from reading the preamble. Therefore, I have dealt with the first paragraph of the preamble and have a full grid. Then it gets more difficult! I think I can picture how the rotating square slices works but there are thousands of options. Has anybody sussed a quick and easy way to proceed?

Thanks

the last paragraph of the preamble is key. the 9 squares with single digit letters initially must add to 200 after the rotations, So, write down the 4 possibilities for each of those squares (one for each 90 degree rotation).

Note that the "1" doesn't move, and one other number is in the middle of a magic square and thus doesn't move either. The constraints put on the other 7 squares let you narrow it down pretty quickly.

...and by "single digit letters" I mean "single digit numbers"

There are fewer options than suggested by kligore trout. For each of the seven cases that change there are three, not four theoretical possibilities, but in each of three cases there are only two possibilities since the third does not satisfy the criteria given in the preamble.
The total sum is even. Since the sum of two primes must be even and the multiples of 4 are even, it follows that the two squares must both be odd or both be even.
There are four possible multiples of 4, of which three are needed, so the there are only four possible totals from those to consider.
With all this in mind, the options to consider are extremely limited. It doesn't take all that long.

Thank you Kilgore Trout and Dryden,

Your nudges and hints have helped me to complete the puzzle. If only I'd noticed that one of the single digit squares is in the middle of a magic square and therefore cannot be affected......!