  We prove two kinds of $\mathbb{Z}/2$-periodic Koszul duality equivalences for
triangulated categories of matrix factorizations associated with $(-1)$-shifted
cotangents over quasi-smooth affine derived schemes. We use this result to
define $\mathbb{Z}/2$-periodic version of Donaldson-Thomas categories for local
surfaces, whose $\mathbb{C}^{\ast}$-equivariant version was introduced and
developed in the author's previous paper. We compare $\mathbb{Z}/2$-periodic DT
category with the $\mathbb{C}^{\ast}$-equivariant one, and deduce wall-crossing
equivalences of $\mathbb{Z}/2$-periodic DT categories from those of
$\mathbb{C}^{\ast}$-equivariant DT categories.
