Let $M$ be a von Neumann algebra. Suppose $v,w$ are partial isometries in $M$ such that $v^*v=w^*w$ and $vv^*=ww^*$. We know that there exist partial isomeries $c_1,c_2$ such that $v=c_1w$ and $v^*=c_2w^*$.
Whether there exists a relationship between $c_1$ and $c_2$?
Write $p=v^*v=w^*w$ and $q=vv^*=ww^*$. On the range of $q$, you necessarily have $$ c_1q=c_1ww^*=vw^*=vw^*q. $$ So $c_1=vw^*$ on the range of $q$. On the range of $1-q$ it can be defined as any partial isometry and the equality $v=c_1w$ will still hold.
Similarly, on the range of $p$ you have $$ c_2p=c_2w^*w=v^*w=v^*wp. $$ That is, $c_2=v^*w$ on the range of of $p$. If we consider the minimal version of $c_1$ and $c_2$, that is we require $c_1(1-q)=0$ and $c_2(1-p)=0$, then $c_1=vw^*$ and $c_2=v^*w$.
If you want to find a relation between, them, you have $$ c_2w^*=v^*=w^*c_1^*, $$ which implies $$ c_2=w^*c_1w. $$ We have $$ c_1^*c_1=wv^*vw^*=wpw^*=q,\qquad c_1c_1^*=vw^*wv^*=vqv^*=q $$ and similarly $$ c_2^*c_2=c_2c_2^*=p. $$