  We establish fundamental motivic results about hermitian K-theory without
assuming that 2 is invertible on the base scheme. In particular, we prove that
both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich
descent, and that symmetric Grothendieck-Witt theory further satisfies
d\'evissage and A^1-invariance over a regular Noetherian base of finite Krull
dimension, as well as a projective bundle formula. We use this to show that
over a regular Noetherian base, symmetric Grothendieck-Witt theory is
represented by a motivic E-infinity-ring spectrum, which we then show is an
absolutely pure spectrum, answering a question of D\'eglise. As with algebraic
K-theory, we show that over a general base, one can also construct a hermitian
K-theory motivic spectrum, representing this time a suitable homotopy invariant
and Karoubi-localising version of Grothendieck-Witt theory.
