  For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an
equivalence $\rho$ between the $\infty$-category of cellular motivic $A$-module
spectra and modules over an $\mathbb{E}_1$-algebra $\Theta$ in $\mathbb{Z}
$-graded spectra, under which the motivic grading corresponds to the
$\mathbb{Z}$-grading.
  If the base is the complex numbers or if $A$ admits an
$\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $\Theta$
to an $\mathbb{E}_\infty$-algebra and $\rho$ to a symmetric monoidal
equivalence. To capture the symmetric monoidal structure in the general
situation, we lift $\rho$ to a symmetric monoidal equivalence to modules over
an $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invert
morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of
Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of
finite sets and bijections.
