In the 80s, Mahowald used the bo-Adams spectral sequence to determine the v1-periodicity in the stable homotopy groups of spheres. Towards computing this spectral sequence, one must compute the ring of cooperations for bo. In motivic homotopy theory, there is a spectrum representing Hermitian K-theory called kq, which has many properties similar to bo. We will compute the ring of cooperations for Hermitian K-theory over the real numbers and show what this indicates for the real kq-Adams spectral sequence. Time permitting, we will discuss related splittings in joint work with Guchuan Li, Sarah Petersen, and Liz Tatum.