For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$.
In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course it's not trivial to do so, but is there a way to do this when $A$ is an $E_2$-algebra, meaning a little disks algebra? In particular, I'm looking at the bar construction as a homotopy G-algebra, as in the first part of http://arxiv.org/pdf/math/0406502v1.pdf.
The idea comes from trying to make $Tor_A(M,N)$ a module, not just a group. It seems that if we have $M$ a right $A$ module and $N$ a left $A$ module, we've used up all the module structure already in computing the homology.
