Does no correlation but dependence imply a symmetry in the joint variable space? I was looking through the answers to this question, and all of them seem to have some form of symmetry between the variables.
I'll walk through the examples in that question so you can see what I mean.
Coin Game Example
$Y = \pm X$ and $X = |Y|$ (symmetrical functions)
The $x^2$ and $x$ Example
Both the uniform distribution and squaring the samples of the uniform distribution are symmetric about the y axis.
Circle Example
Circles are symmetrical across both X and Y axes. (Can include many other axes as well, but I'm not 100% that if you chose different axes that you would get no correlation, so I'm just going to say X and Y.)
Car Velocity Example
Similar to $x$ and $x^2$ case, $K$ is essentially $V^2$, and $V$ is essentially $\pm \sqrt{K}$. Again, both functions are symmetric.
So, does 0 correlation and dependence imply a symmetry in the joint variable space? Can anyone describe the symmetry mathematically or provide a mathematical proof that there is a symmetry in the joint space?

NO
Consider the following bivariate distribution, with uniform probability across the six points.
$$
\{
(1,1),(2,2),(3,3),(4,4),(5,5),(2,15)
\}
$$
The correlation is zero. However, $P(Y=5)=\frac{1}{6}$ while $P(Y=5\vert X=5)=1$, so the variables are dependent. At the same time, I see no obvious symmetry in the bivariate distribution when I graph it.


Proof that the correlation is zero
Correlation is zero if the covariance between the variables is zero.
$$Cov(X, Y) = \mathbb E[XY] - \mathbb E[X]\mathbb E[Y]$$
$$\mathbb E[XY] = \dfrac{1}{6}\left((1\times1) + (2\times2) + (3\times3)+(4\times4) + (5\times5)+(15\times 2)\right) = \dfrac{85}{6}$$
$$\mathbb E[X] = \dfrac{1}{6}(1+ 2+ 3+ 4+ 5+ 2) = \dfrac{17}{6}$$
$$\mathbb E[Y] = \dfrac{1}{6}(1 + 2 + 3 + 4 + 5 + 15) = 5$$
$$\implies$$
$$Cov(X, Y) = \dfrac{85}{6} - \left(\dfrac{17}{6} \times 5\right) = 0$$
Thus, the variables are uncorrelated.
# Code for the plot with the dotted lines

library(ggplot2)
set.seed(2022)
x <- c(1, 2, 3, 4, 5)
y <- c(1, 2, 3, 4, 5)
x <- c(x, 2)
y <- c(y, 15)
cor(x, y)
plot(x, y)
sqrt((1 - 2)^2 + (1-15)^2)
sqrt((5 - 2)^2 + (5-15)^2)
d <- data.frame(X = x, Y = y)
e <- data.frame(x = c(1, 2, 5, 1), y = c(1, 15, 5, 1))
f <- data.frame(x = c(1, 5), y = c(1, 5))
p <- ggplot(d, aes(x = X, y = Y)) + 
  geom_point() + 
  scale_x_continuous(limits = c(0, 16)) +
  scale_y_continuous(limits = c(0, 16))
p <- p + geom_line(data = e, aes(x = x, y = y),linetype = "dotted") 
p <- p + geom_line(data = f, aes(x = x, y = y),linetype = "dotted")
# p <- p + geom_line(data = g, aes(x = x, y = y),linetype = "dotted")
p# + theme_bw()

