  The "$10$-fold way" refers to the combined classification of the $3$
associative division algebras (of real, complex and quaternionic numbers) and
of the $7$, ${\mathbb Z}_2$-graded, superdivision algebras (in a superdivision
algebra each homogeneous element is invertible). The connection of the
$10$-fold way with the periodic table of topological insulators and
superconductors is well known. Motivated by the recent interest in ${\mathbb
Z}_2\times{\mathbb Z}_2$-graded physics (classical and quantum invariant
models, parastatistics) we classify the associative ${\mathbb Z}_2\times
{\mathbb Z}_2$-graded superdivision algebras and show that $13$ inequivalent
cases have to be added to the $10$-fold way. Our scheme is based on the
"alphabetic presentation of Clifford algebras", here extended to graded
superdivision algebras. The generators are expressed as equal-length words in a
$4$-letter alphabet (the letters encode a basis of invertible $2\times 2$ real
matrices and in each word the symbol of tensor product is skipped). The $13$
inequivalent ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebras
are split into real series ($4$ subcases with $4$ generators each), complex
series ($5$ subcases with $8$ generators) and quaternionic series ($4$ subcases
with $16$ generators).
