We examine some of the roots of parity violation for gravitons and uncover a closely related new effect: correlations between right- and left-handed gravitons. Such correlators have spin 4 if they involve gravitons moving along the same direction and spin zero for gravitons moving with opposite directions. In the first case, the most immediate implication would be a degree of linear polarization for the tensor vacuum fluctuations, which could be seen by gravity-wave detectors sensitive enough to probe the primordial background, its degree of polarization and anisotropies. Looking at the anisotropy of the gravity waves linear polarization, we identify the parity respecting and violating components of the effect. The imprint on the cosmic microwave background temperature and polarization would be more elusive, since it averages to zero in the two-point functions, appearing only in their cosmic variance or in fourth-order correlators. In contrast, spin-zero correlations would have an effect on the two-point function of the cosmic microwave background temperature and polarization, enhancing the BB component if they were anticorrelations. Such correlations represent an amplitude for the production of standing waves, as first envisaged by Grishchuk, and could also leave an interesting signature for gravity-wave detectors.