Cross-Correlating Astrophysical and Cosmological Gravitational Wave Backgrounds with the Cosmic Microwave Background

General relativity provides us with an extremely powerful tool to extract at the same time astrophysical and cosmological information from the stochastic gravitational-wave backgrounds (SGWBs): the cross-correlation with other cosmological tracers, since their anisotropies share a common origin and the same perturbed geodesics. In this Letter we explore the cross-correlation of the cosmological and astrophysical SGWBs with cosmic microwave background (CMB) anisotropies, showing that future GW detectors, such as LISA or BBO, have the ability to measure such cross-correlation signals. We also present, as a new tool in this context, constrained realization maps of the SGWBs extracted from the high-resolution CMB Planck maps. This technique allows, in the low-noise regime, to faithfully reconstruct the expected SGWB map by starting from CMB measurements.