We consider the state complexity of projection, the shuffle operation, intersection, union, up- and downward closure and interior on commutative regular languages. Usually, state complexity bounds are given in terms of the state complexities of the input languages (the sharp bound nm for union and intersection on commutative, even unary, languages of state complexities n and m, respectively, is known in the literature). Here, we associate with every commutative regular language an index and period vector, generalizing corresponding notions for unary and periodic languages from the literature, and express sharp state complexity bounds with these novel parameters. Using relations between the state complexity of a language and the index and period vectors, we deduce the state complexity bound n|Σ| for upward closure and downward interior, and (2nm)|Σ|, (nm)|Σ| and (n+m−1)|Σ| for the shuffle of general commutative regular, commutative group and commutative aperiodic languages, respectively, with state complexities n and m over an alphabet Σ. We do not know whether these bounds are sharp. We also prove the sharp bound n for projections and downward closure and upward interior. Our results are based on a canonical automaton model for commutative languages. We also show that commutative regular languages can be written as a finite union of shuffle products of commutative finite and commutative group languages intersected with a subalphabet. Furthermore, we prove characterizations of the commutative aperiodic and commutative group languages in terms of the index and periodic vectors and investigate relations between a language and their unary projection languages.