I am reading Kirk and Davis's lecture notes and on page 99 they claim that the singular chain complex of the universal cover of a space is a right $\mathbb{Z}\pi$ module.
I believe they use the convention that the composition of functions $f \circ g$ is the function $x \rightarrow f(g(x))$ and similarly composition of loops $\alpha * \beta$ is given by traversing $\beta$ then $\alpha$.
The action makes sense to me: if $\gamma \in \pi_1(X)$ and $\sigma \in C^n(\bar{X})$ take $\sigma$ to $\gamma \circ \sigma$ where we identify $\gamma$ with its deck transformation.
To me it seems like this is a left module since if $\alpha,\beta \in \pi_1(X)$, we have $(\alpha \circ \beta)\sigma=\alpha \circ \beta \circ \sigma=\alpha(\beta \sigma)$.
I have thought about the identification of the fundamental group with the deck transformation group, and I do not think that is the issue.