at.algebraic topology - Commutativity in K-theory and cohomology

Perhaps the deeper story you want involves the notion of "E-infinity product". The cup product in cohomology, and the sum (and for that matter, the product) in K-theory are commutative (and associative, and unital) not merely up to homotopy, but "up to all possible higher homotopies". You can make this precise by saying that the appropriate binary operation on the representing space (the K(Z,n)'s, or Z x BU), is part of an E-infinity algebra structure on that space.

It seems that most "naturally occuring" sums or products in topology turn out to be E-infinity (addition for any generalized cohomology theory, multiplication for many nice ones such as ordinary cohomology, K-theories, bordism, elliptic cohomology). As you observe, having a strictly commutative operation is very special, and basically forces the representing spaces to be a product of K(A,n)'s, by the Dold-Thom theorem.

To me, the mystery here is that "stricly commutative" is so much more special than "E-infinity commutative". Associative products don't behave this way: "strictly associative" turns out to be no more special than "A-infinity associative", that is, any A-infinity product on a space can be "rigidified" to a strictly associative product on a weakly equivalent space.