  The anisotropies of the stochastic gravitational wave background, as produced
in the early phases of cosmological evolution, can act as a key probe of the
primordial universe particle content. We point out a new universal property of
gravitational wave anisotropies of cosmological origin: for adiabatic initial
conditions, their angular power spectrum is insensitive to the equation of
state of the cosmic fluid driving the expansion before big-bang
nucleosynthesis. Any deviation from this universal behaviour points to the
presence of non-adiabatic sources of primordial fluctuations. Such scenarios
can be tested by gravitational wave detectors operating at a frequency range
which is fully complementary to CMB experiments. In this work we prove this
general result, and we illustrate its consequences for a representative
realisation of initial conditions based on the curvaton scenario. In the case
of the simplest curvaton setup, we also find a significant cross-correlation
between gravitational wave anisotropies and the CMB temperature fluctuations.
There is a fourfold enhancement vis-\`{a}-vis the purely adiabatic scenario. We
discuss the implications of our findings for identifying the origin of the
(cosmological) gravitational wave background when, as is often the case, this
cannot be determined solely on the basis of its spectral shape.
