  We show that the tensor product of two cyclic $A_\infty$-algebras is, in
general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with
homotopy inner product. More precisely, we construct an explicit combinatorial
diagonal on the pairahedra, which are contractible polytopes controlling the
combinatorial structure of an $A_\infty$-algebra with homotopy inner products,
and use it to define a categorically closed tensor product. A cyclic
$A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy
inner products whose higher inner products are trivial. However, the higher
inner products on the tensor product of cyclic $A_\infty$-algebras are not
necessarily trivial.
