Symmetry conditions for symmetric random vectors While formulating the properties for a certain statistical model I'm dealing with, I came up with the following question (with credit going to MikeEarnest in comments for the proper formulation). A random variable $X$ is symmetric (about zero) if $X$ and $-X$ are identically distributed. This in particular applies to the case of a random vector. The question is then:

Suppose $X=(X_1,X_2,\ldots, X_n)$ is a random vector such that $(X_{i(1)},X_{i(2)},…,X_{i(m)})$ is symmetric for any $\{i(1),i(2),…,i(m)\}\subseteq \{1,2,…,n\}$. Does it follow that $X$ is symmetric?

This statement is probably too generic, given the lack of assumptions on $X$ beyond symmetry. If I specialize to the case where each $X_k$ can only take values $\pm 1$, then my understanding is that the above is true when $m=1$ regardless of $n$. (This would imply that it is true for arbitrary $m$, since the subsets of $\{1,2,\cdots, n\}$ of size at most $m$ includes the singletons.)  However, if I move to $X_k$ taking values in $\{1,0,-1\}$ then the case of $m=1$ fails (see the comments for a counter-example due to MikeEarnest) and I don't know what can be said about $m>1$.
This gives me the following (admittedly imprecise) question: What assumptions on the maximum size $m$ and the distribution of the random variables $\{X_k\}_{k=1}^n$ must be made in order for the above statement to be valid? I would also be interested in  additional counter-examples.
 A: The statement is untrue even for $m=n-1$. There exists an asymmetric random vector such that any $n-1$ coordinates form a symmetric vector. An example is $\newcommand{\b}[1]{{\bf #1}}\b X = (X_1,X_2,\dots,X_n)$ with support $\{-1,0,1\}^n$ and the following joint pmf:
$$
\mathbb P\big(\b X=(x_1,x_2,\dots,x_n)\big)=
\begin{cases}
3^{-n} & \text{if any }x_i=-1\\
3^{-n}(1+(-1)^{x_1+\dots+x_n}) & \text{if all }x_i\in \{0,1\}
\end{cases}
$$
You can show that show that any sublist of $(X_1,\dots,X_n)$ is symmetric. More specifically, a sublist of length $m<n$ is uniform on $\{-1,0,1\}^m$. However, $\b X$ is not symmetric, as $$P\big (\b X=(0,0,\dots,0,1)\big)=0\neq 3^{-n}=P\big(\b X=(0,0,\dots,0,-1)\big).$$
For example, when $n=2$, the joint distribution is
$$
\begin{array}{cc} & X_2\\ X_1 &\begin{array}{c|cc} &1&0&-1\\ \hline 1 &2/9 & 0 & 1/9\\ 0 &0 & 2/9 & 1/9\\ -1 &1/9 & 1/9 & 1/9 \end{array} \end{array} 
$$
while both $X_1$ and $X_2$ are uniform on $\{-1,0,1\}$. 
Another example where $n=3$:
$$
\begin{array}{|lr|ccc|ccc|ccc|} 
\hline
X_3& & & 1 & & &0& & &-1& &\\
\hline
X_2& & 1 & 0 & -1& 1 & 0 & -1& 1 & 0 & -1\\ 
\hline
&1  & 0&2/27 & 1/27&2/27 & 0 & 1/27&1/27 & 1/27 & 1/27\\ 
 X_1& 0&2/27 & 0 & 1/27 & 0&2/27 & 1/27&1/27 & 1/27 & 1/27\\ 
 &-1&1/27 & 1/27 & 1/27&1/27 & 1/27 & 1/27&1/27 & 1/27 & 1/27\\
\hline
\end{array} 
$$
