  In this paper we begin the study of the (dual) Steenrod algebra of the
motivic Witt cohomology spectrum $H_W\mathbb{Z}$ by determining the algebra
structure of ${H_W\mathbb{Z}}_{**}H_W\mathbb{Z}$ over fields $k$ of
characteristic not $2$ which are extensions of fields $F$ with $K^M_2(F)/2=0$.
For example, this includes all fields of odd characteristic, as well as fields
that are extensions of quadratically closed fields of characteristic $0$. After
inverting $\eta$, this computes the $HW:=H_W\mathbb{Z}[\eta^{-1}]$-algebra
${HW}_{**}HW$. In particular, for the given base fields, this implies the
$HW$-module structure of $HW\wedge HW$ recently computed by Bachmann and
Hopkins.
