I would like your help in order to choose the appropriate terminology for classifying the assumptions below.
Consider a random vector $X\equiv (X_1,X_2,X_3)$.
Let $\Delta X\equiv (X_1-X_3, X_2-X_3, X_1-X_2)$ with probability distribution denoted by $P_{\Delta X}$ whose marginals are $(P_{\Delta X})_k$ for $k\in \{1,2,3\}$.
Take the following possible assumptions on $P_{\Delta X}$:
(1) $(P_{\Delta X})_1=(P_{\Delta X})_2=(P_{\Delta X})_3$
(2) $(P_{\Delta X})_k$ has median zero $\forall k\in \{1,2,3\}$
(3) $(P_{\Delta X})_k$ is symmetric around zero $\forall k\in \{1,2,3\}$
Assumptions (1), (2), (3) are all implications of assuming
(4) $\{X_1,X_2,X_3\}$ are exchangeable.
Question: I would like your help to find a generic and common way to refer to assumptions (1), (2), (3).
At the moment I thought about
"(1), (2), (3) are assumptions imposing various degrees of symmetry on $P_{\Delta X}$"
motivated by the fact that (1), (2), (3) are implications of exchangeability which is indeed an assumption of symmetry. However, I don't like this taxonomy very much because of the word "symmetry" which may confuse the reader since it appears also in (3).
Also, I don't want to explicitly mention the fact that (1), (2), (3) are implications of exchangeability, such as something like
"(1), (2), (3) are different implications of exchangeability"
Any better suggestion?