  Over any field of characteristic not 2, we establish a 2-term resolution of
the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the
connective 2-local Witt K-theory spectrum. This is curiously similar to the
resolution of the K(1)-local sphere in classical stable homotopy theory. As
applications we determine the $\eta$-periodized motivic stable stems and the
$\eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way
we construct Adams operations on the motivic spectrum representing Hermitian
K-theory and establish new completeness results for certain motivic spectra
over fields of finite virtual 2-cohomological dimension. In an appendix, we
supply a new proof of the homotopy fixed point theorem for the Hermitian
K-theory of fields.
