In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group $${\text {Mod}}_g$$ and the Torelli group $$\mathcal {I}_g$$ . We show that Kawazumi’s twisted MMM class $$m_{0,k}$$ can be used to compute k-fold cup products in surface bundles, and that $$m_{0,k}$$ provides an extension of the higher Johnson invariant $$\tau _{k-2}$$ to $$H^{k-2}({\text {Mod}}_{g,*}, \wedge ^k H_1)$$ . These results are used to show that the behavior of the restriction of the even MMM classes $$e_{2i}$$ to $$H^{4i}(\mathcal {I}_g^1)$$ is completely determined by $${\text {im}}(\tau _{4i}) \le \wedge ^{4i+2}H_1$$ , and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel $$\mathcal K_{g,*}$$ have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all $$\tau _i$$ when restricted to $$\mathcal K_{g,*}$$ .