Can finite dimensional algebras (over a field $K$) A with $D(A) \otimes_A D(A) \cong A$ as $A$-bimodules be somehow classified? I think taking for A a selfinjective algebra with nakayama permutation of order at most two should work. Are there nonselfinjective examples? Here $D(A)=Hom_K(A,K)$ is the dual of the regular module. Here a little motivation:

Symmetric algebras are characterised by $D(A) \cong A$ as bimodules.

More generally define the cyclic monoid generated by $D(A)$ with multiplication $\otimes_A$, when is this a group? That is, which algebras satisfy $A \cong D(A)^{\otimes i}$ for some $i \geq 1$? Are there non-selfinjective examples? It might also be interesting to look just at one-sided isomorphisms instead of bimodule isomorphisms, where this notion might generalise frobenius algebras.