Choose some basis in $\mathbb{R}^n$ and call one of its directions as "$z$". Let $x$ be a $n-$dimensional random variable s.t for all $3-$tuple of $n-$dimensional rotation matrices $R_1, R_2$ and $R_3$ I want the following expectation to be a constant (independent of $R_1, R_2$ and $R_3$),
$$\mathbb{E}_{x} \left[ (R_1 x)_z \vert (R_2 x)_z \vert \vert (R_3 x)_z \vert \right]$$
- Are there non-trivial distributions with such properties? (...trivial cases are when the above is $0$ for the distribution being parity symmetric..)
Feel free to assume reasonable restrictions on the support of the distribution - like if say compact support on a sphere or such helps!