  We study a series of real nonassociative algebras $\mathbb{O}_{p,q}$
introduced in $[5]$. These algebras have a natural $\mathbb{Z}_2^n$-grading,
where $n=p+q$, and they are characterized by a cubic form over the field
$\mathbb{Z}_2$. We establish all the possible isomorphisms between the algebras
$\mathbb{O}_{p,q}$ preserving the structure of $\mathbb{Z}_2^n$-graded algebra.
The classification table of $\mathbb{O}_{p,q}$ is quite similar to that of the
real Clifford algebras $\mathrm{Cl}_{p,q}$, the main difference is that the
algebras $\mathbb{O}_{n,0}$ and $\mathbb{O}_{0,n}$ are exceptional.
