If $u\cdot v \in (z \cdot z')^*$ and $v \cdot u \in (z' \cdot z)^*$ then $u \in (z \cdot z')^* \cdot z$
The full statement (including the last point) seems to follow directly from the proof of Proposition 1.3.3 in (Lothaire '83).  The last point is part of the proof: using the notation in the original post, the proof states that $u = (z \cdot z')^{k_1} \cdot z$ and $v = z' \cdot (z \cdot z')^{k_2}$ for some $k_1$ and $k_2$ such that $k_1+k_2+1 = c$.  Perhaps the proof of this proposition could be cited...?