In this paper, we introduce cell-forms on $\mathcal{M}_{0,n}$, which are
top-dimensional differential forms diverging along the boundary of exactly one
cell (connected component) of the real moduli space
$\mathcal{M}_{0,n}(\mathbb{R})$. We show that the cell-forms generate the
top-dimensional cohomology group of $\mathcal{M}_{0,n}$, so that there is a
natural duality between cells and cell-forms. In the heart of the paper, we
determine an explicit basis for the subspace of differential forms which
converge along a given cell $X$. The elements of this basis are called
insertion forms, their integrals over $X$ are real numbers, called cell-zeta
values, which generate a $\mathbb{Q}$-algebra called the cell-zeta algebra. By
a result of F. Brown, the cell-zeta algebra is equal to the algebra of
multizeta values. The cell-zeta values satisfy a family of simple quadratic
relations coming from the geometry of moduli spaces, which leads to a natural
definition of a formal version of the cell-zeta algebra, conjecturally
isomorphic to the formal multizeta algebra defined by the much-studied double
shuffle relations.