Does commutativity extend to functions with 3+ arguments? As far as I know, commutativity is a property of binary operators. 
Does it apply to functions with 3+ arguments? If so, are all permutations considered, or just rotations, or what?
 A: Considering a binary operation as a function of its two operands, one way to extend the idea of commutativity to more than two operands is to consider symmetric functions.
The value of a symmetric function applied to some list of parameters is the same as its value for any permutation of that list of parameters.
If an operation on $n$ operands is defined as a symmetric function of the operands,
then by definition its value does not change due to any permutation of the operands. 
You could also consider a restricted set of permutations of the parameters. Your suggestion of allowing rotations could be interesting. 
Another possibility is to consider an operation whose result is unchanged by any even permutation of its operands (that is, a permutation that can be implemented by an even number of transpositions; a transposition swaps the positions of two operands).
A special case of allowing even permutations of the parameters is an operation whose result is negated whenever you transpose two of its operands. 
