Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$

Let $M$ be a symmetric matrix over $\mathbb{F}_2$, and let $\operatorname{diag}(M)$ be its diagonal vector. It is known that $\operatorname{diag}(M)\in\operatorname{Im}(M)$. Thus the affine system $Mx=\operatorname{diag}(M)$ is always solvable. We strengthen this existence statement to a parity rigi