In $Z[x]$, which pairs of $a,b$ commute as modulo operators, such that ($c$ mod $a$) mod $b =$ ($c$ mod $b$) mod $a$ for every $c$?

Question

In $Z[x]$, which pairs of $a,b$ commute as modulo operators, such that ($c$ mod $a$) mod $b =$ ($c$ mod $b$) mod $a$ for every $c$?

What I got so far is:

Clearly the equasion holds for every pair where one polynomial divides the other.

Another group of pairs are these with $deg(a)=0$ and lead_coeff$(b)=1$.

However, especially the second group seems pretty arbitrary, and it feels like I am missing the bigger picture. Is there a neccessary and sufficient condition for $a,b$?