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I'm not seeing anything explicit about whether a king sprials clockwise or counter-clockwise (or both.) And does black spiral the opposite way? (Based on your description of king vs king endgames, I'm thinking not, although it seems to me that should be the case - unless, of course, both can spiral in both directions.)
Each King can move along all eight congruent spirals. I had expected that to be clear from calling it a 'fully symmetric' spiral mover (and the earlier explanation that spiral moves are oblique and have 8 realizations). I have added a sentence now to make that more explicit.
By scripting a special move generator on this page I could include an Interactive Diagram that can play this variant.
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