We show that quadratic and symmetric (Formula presented.) -theory of the integers are related by Anderson duality and that both spectra split integrally into the (Formula presented.) -theory of the real numbers and a generalised Eilenberg–Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space (Formula presented.). Finally, we prove analogous results for the genuine L-spectra recently devised for the study of Grothendieck–Witt theory.