The algebraic gadget is: it's got the multiplicative structure of a cyclic group of n things (just like a finite field of n things would) and an additive structure which distributes over the multiplication (and has a unit and is commutative) but isn't necessarily associative (!!). Do these have a standard name? Am I fooling myself that any of these exist that aren't fields? Anyway, this includes all fields at least.

The combinatorial object is: take n dots in a circle, and choose one of them. And choose two of them that are adjacent (i.e. 1 apart) on the circle. And choose two of them that are 2 apart from each other on the circle. And choose two that are 3 apart, etc. until there's only one or zero dots left.

For an example, consider GF(8). Pick a generator g. We know g⁷ = 1.

1 + g has to equal g ( 1 + g⁶ ) = g + 1 by commutativity and distributivity of addition.
1 + g² has to equal g² ( 1 + g⁵ ) in the same way

So if we write out the first row of a possible addition-table for GF(8)

Those are the choices of pairs on the necklace. So it's possible to make a necklace-with-pairings on a pentagon, in exactly one way: if the pentagon is ABCDE, choose A, and the pair CD (1 apart) and the pair BE (2 apart). And this seems to give rise to some distributive algebraic structure like I said above, but its addition isn't associative --- and it couldn't possibly be, because there's no such thing as GF(6).