We model the stellar density and the rotation curve of the Milky Way and compute the gravitational-wave spectrum of its unresolved binaries as a function of direction. Because the Galaxy rotates, the waves emitted along each line of sight are Doppler-shifted by a different amount, and we derive for the first time the resulting observed anisotropic spectrum, which no longer factorizes into a spectral shape times an angular template. We then quantify the observability of this distortion with space-based interferometers. With their nominal noise budgets, neither LISA nor a LISA+Taiji network can detect it: the residual between the models that include and neglect the rotation has a signal-to-noise ratio ${\rm SNR}_{\rm diff}\simeq0.2$ and $0.7$ after two years, and the induced bias on the foreground parameters stays below $0.1\sigma$. A tenfold better noise amplitude, of the order contemplated for post-LISA concepts, raises the residual to ${\rm SNR}_{\rm diff}\simeq4.6$ for the network, where the effect becomes measurable and displaces the spectral cutoff by half of the statistical error.