Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following questions:
Are there necessary and sufficient conditions on the cumulative distribution function (CDF) associated with $P$ (joint or marginal) ensuring that $$ \exists \text{ a random vector $(X_0,X_1,X_2)$ such that } $$ $$ (X_1-X_0, X_1-X_2), (X_2-X_0, X_2-X_1), (X_0-X_1, X_0-X_2) $$ $$ \text{ have all probability distribution $P$? } $$
Notice:
$(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate. For example, $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.
My thoughts: among the necessary conditions, I would list the following: let $G$ be the CDF associated with $P$ and let $G_1,G_2$ be the two marginal CDFs. Then it should be that $$ \begin{cases} G_1 \text{ is symmetric around zero, i.e., $G_1(a)=1-G_1(-a)$ $\forall a \in \mathbb{R}$}\\ G_2 \text{ is symmetric around zero, i.e., $G_2(a)=1-G_2(-a)$ $\forall a \in \mathbb{R}$}\\ \end{cases} $$
Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?