quantum groups - How does one think about the "off-diagonal" part of the $R$-matrix?

Another way to think about this, which I learned from Mark Haiman's class, is that there are actually $4$ natural coproducts on the quantum group. $Deltaleft(Eright)$ could be $Eotimes K + 1 otimes E$, or you could move the $K$ to the other factor, or you could invert $K$, or you could do both. The individual parts of the $R$-matrix move between these four different coproducts. In particular, the quasi-$R$-matrix (iirc) turns the $K$ into a $K^{-1}$.