From what I've read, there is a statement that is roughly like:
"If a sequence of random variables is IID conditional on some other random variables, then it is exchangeable",
but I can't find a rigorous trustworthy source from which to just cite this result or whatever the correct version of it is (it isn't even 100% clear what it means in the first place - there is more than one possible interpretation).
The interpretation that I currently think is most likely to be the correct one is the following... Is the following statement true?
Let $(X_j)_{j=1}^{\infty} : (\Omega,\mathscr{F},\mathbf{P}) \to (\mathbf{R},\mathscr{B}(\mathbf{R}))$ be a sequence of random variables. If there is another random variable $Y$ such that for every $N \geq 1$ we have: $$ \mathbf{E}\bigl[ X_1\cdots X_N | \sigma(Y)\bigr] = \mathbf{E}\bigl[X_1 | \sigma(Y)\bigr]^N, $$ then the sequence $(X_j)_{j=1}^{\infty}$ is exchangeable.