Setup: Let $X_t$ and $Y_t$ denote two (possibly dependent) random variables with cumulative distribution functions (cdf) $F_X$ and $F_Y$. Assume the support of $F_X$ and $F_Y$ is $\mathbb{R}^+$. Let $Z_t = X_t - Y_t$, and let $F_Z$ denote the cdf of $Z_t$.
Question: What are the necessary conditions that must be placed on $F_X$ and $F_Y$ to ensure that $F_Z$ is symmetric, or approximately symmetric, around its first moment? Any references on this question would also be greatly appreciated. Edit: By symmetry, I mean that if $\mu = \mathbb{E} Z_t$, then $F_z(\mu - c) = 1 - F_z(\mu + c), \forall c$. Equivalently, if $f_Z$ denotes the probability distribution function, then $f_Z(\mu - c) = f_Z(\mu + c), \forall c$.
Motivation: One known result is that if $X_t$ and $Y_t$ are independent and identically distributed according to the exponential distribution, then $Z_t$ will follow the Laplace distribution, which is symmetric around its first moment (which is zero). I was wondering if there are some more general conditions under which $Z_t$ will follow a symmetric distribution.
If $X$ and $Y$ are i.i.d. (not necessarily exponential) then the distribution of $Z$ is symmetric. If $X$ and $Y$ are allowed to be dependent, the situation becomes horribly complicated.