I can't say I understand your notion of "morphism of quiver representations," but I can tell you a canonical bijection between representations for one orientation and another, which might be what you're looking for.

Let $\bar Q$ be the doubled $A_n$ quiver, that is, the same set of dots with all arrows replaced by a pair of oppositely oriented arrows.  

Given a representation $M$ of $Q$, consider the representations $E_M$ of $\bar Q$ on the same vector space that agree with $M$ on the arrows in $Q$, do whatever they want on the others, and have the property that the paths $i\to i-1 \to i$ and $i\to i+1\to i$ give the same self-map of $M_i$.  Note that this is a set of *linear* equations on the choices for each new arrow in $\bar Q$, so its set of solutions is a linear space.

Now, consider the map of this linear space to the set of $Q'$-representations given by forgetting the arrows not in $Q'$.  This isn't independent of our choices, but there's one isomorphism type that occurs generically in $E_M$ and all the others only show up in lower dimensional subvarieties
So, consider the map $\varphi$ that sends a isomorphism class of $Q$ reps to the $Q'$ rep that occurs generically in $E_M$ after forgetting the arrows not in $Q'$.

*Theorem.* The map $\varphi$ is a bijection.  

Note that this bijection is not extremely easy to describe combinatorially;  for example, it is *piecewise* linear in the multiplicity of the different indecomposables in your representation.