The category of $\mathbb{Z}$-graded abelian groups is equivalent to the category of comodules over the commutative Hopf algebra (over $\mathbb{Z}$) $A=\mathbb{Z}[t,t^{-1}]$, with comultiplication $t\mapsto t\otimes t$ and counit $t\mapsto 1$. Explicitly, the correspondence is given as follows: given an $A$-comodule $M$ with structure map $\psi:M\to A\otimes M$, let $M_n=\{x\in M:\psi(x)=t^n\otimes x\}$. Then the coassociativity and counitality of $\psi$ can be easily checked to imply that $M$ is actually the direct sum of these $M_n$, and it can similarly be checked that a map $M\to M'$ of two such comodules commutes with the coaction of $A$ iff it maps $M_n$ to $M_n'$ for all $n$. The Hopf algebra $A$ is the ring of functions on the algebraic group $\mathbb{G}_m$, and this equivalence of categories is closely related to the correspondence between graded algebras and projective varieties (through the fact that projective varieties are quotients of certain quasiaffine varieties under an action of $\mathbb{G}_m$).

The commutativity of $A$ gives a natural symmetric monoidal structure on the category of $A$-comodules, given by $M\otimes_{\mathbb{Z}}M'$ with coaction $M\otimes M'\to(A\otimes M)\otimes(A\otimes M')\cong(A\otimes A)\otimes(M\otimes M')\to A\otimes M\otimes M'$, where the first map is given by the coactions on $M$ and $M'$ and the last map is given by the multiplication on $A$. This tensor product corresponds to the standard tensor product of graded modules, in which $(M\otimes M')_n=\bigoplus_{i+j=n}M_i\otimes M_j'$.

In this symmetric monoidal structure (whose existence only depends on the fact that $A$ is a commutative Hopf algebra), the swap isomorphism $M\otimes M'\to M'\otimes M$ is the standard swap map coming from the underlying symmetric monoidal tensor product of $\mathbb{Z}$-modules. This corresponds to the "even" symmetric monoidal structure on graded objects, in which the swap map comes from the standard swap map $M_i\otimes M_j'\to M_j'\otimes M_i$. Commutative monoid objects under this structure are what an algebraic geometer might call graded commutative rings. However, there is a second symmetric monoidal structure on graded abelian groups with the same tensor product but for which the swap map $M_i\otimes M_j'\to M_j'\otimes M_i$ is multiplied by a sign $(-1)^{ij}$. In this symmetric monoidal structure, a commutative monoid object is a "skew-commutative graded ring", or what a topologist would just call a graded commutative ring: even degree elements commute with everything, and odd degree elements anticommute with each other.

My question is: is there a natural interpretation of this second symmetric monoidal structure when we view graded abelian groups as $A$-comodules? Morally, I would say it can't come naturally just from the Hopf algebra structure of $A$, because we could reinterpret $A$ as the ring $\mathbb{Z}[s^2,s^{-2}]$ so that everything ought to be treated as having even grading, and then we get the original symmetric monoidal structure (this is like reinterpreting a $\mathbb{Z}$-graded ojbect as actually being $2\mathbb{Z}$-graded). Thus, I suppose I'm really asking: is there some additional structure you can put on $A$ that makes the "anticommutative" symmetric monoidal structure pop out naturally (and have a purely categorical description)?