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We show that the generalized Khovanov homology, defined by the second authorin the framework of chronological cobordisms, admits a grading by the group$\mathbb{Z}\times\mathbb{Z}_2$, in which all homogeneous summands areisomorphic to the unified Khovanov homology defined over the ring$\mathbb{Z}_{\pi}:=\mathbb{Z}[\pi]/(\pi^2-1)$ (here, setting $\pi$ to $\pm 1$results either in even or odd Khovanov homology). The generalized homology has$\Bbbk := \mathbb{Z}[X,Y,Z^{\pm 1}]/(X^2=Y^2=1)$ as coefficients, and the aboveimplies that most of automorphisms of $\Bbbk$ fix the isomorphism class of thegeneralized homology regarded as $\Bbbk$-modules, so that the even and oddKhovanov homology are the only two specializations of the invariant. Inparticular, switching $X$ with $Y$ induces a derived isomorphism between thegeneralized Khovanov homology of a link $L$ with its dual version, i.e. thehomology of the mirror image $L^!$, and we compute an explicit formula for thismap. When specialized to integers it descends to a duality isomorphism for oddKhovanov homology, which was conjectured by A. Shumakovitch.

Mirror links have dual odd and generalized Khovanov homology