If two random variables are identically distributed (but not necessarily independent), are they exchangeable?

Question

If $X$ and $Y$ are identically distributed random variables, but not necessarily independent, are they exchangeable?

$P(X = a, Y = b) = P(X=b, Y=a)$?

Answer 1

No. You have to go to more than a binary random variable, but you can find a counterexample with three values. Consider a joint distribution like $P(X=a,Y=b) =M_{ab}$ with $$ M=\begin{pmatrix} p & q & r \\ r & p & q \\ q & r & 1-2p-2q-2r\end{pmatrix}$$ where $2p+2q+2r < 1$ and $p\ne q\ne r.$