Suspensions of rear- and front-actuated microswimmers immersed in a fluid, known respectively as "pushers" and "pullers," display qualitatively different collective behaviors: beyond a characteristic density, pusher suspensions exhibit a hydrodynamic instability leading to collective motion known as active turbulence, a phenomenon which is absent for pullers. In this Letter, we describe the collective dynamics of a binary pusher-puller mixture using kinetic theory and large-scale particle-resolved simulations. We derive and verify an instability criterion, showing that the critical density for active turbulence moves to higher values as the fraction χ of pullers is increased and disappears for χ≥0.5. We then show analytically and numerically that the two-point hydrodynamic correlations of the 1∶1 mixture are equal to those of a suspension of noninteracting swimmers. Strikingly, our numerical analysis furthermore shows that the full probability distribution of the fluid velocity fluctuations collapses onto the one of a noninteracting system at the same density, where swimmer-swimmer correlations are strictly absent. Our results thus indicate that the fluid velocity fluctuations in 1∶1 pusher-puller mixtures are exactly equal to those of the corresponding noninteracting suspension at any density, a surprising cancellation with no counterpart in equilibrium long-range interacting systems.