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We establish a fibre sequence relating the classical Grothendieck-Witt theoryof a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory andRanicki's original (non-periodic) symmetric L-theory. We use this fibresequence to remove the assumption that 2 is a unit in $R$ from various resultsabout Grothendieck-Witt groups. For instance, we solve the homotopy limitproblem for Dedekind rings whose fraction field is a number field, calculatethe various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that theGrothendieck-Witt groups of rings of integers in number fields are finitelygenerated, and that the comparison map from quadratic to symmetricGrothendieck-Witt theory of Noetherian rings of global dimension $d$ is anequivalence in degrees $\geq d+3$. As an important tool, we establish thehermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekindrings and use it to solve a conjecture of Berrick-Karoubi.

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings