The well-known reflection formula for the Gamma function relates its values at $z$ and $1-z$ for any $z\in\mathbb{C}$, and has the explicit form
\begin{equation} \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{\pi z}}. \end{equation}
Is there an appropriate generalization of this formula to the case of a triple product? In particular, is there some simplified form of the expression
\begin{equation} \Gamma(x+y-z)\,\Gamma(x+z-y)\,\Gamma(y+z-x)? \end{equation}
In the context of a certain calculation in conformal field theory, there are physical reasons to believe that, for $x,y,z\in\mathbb{R}$, this expression should factor into a function of the form
\begin{equation} C(\{x\},\{y\},\{z\})f(x)f(y)f(z), \end{equation}
where $\{\cdot\}$ denotes the fractional part, and $f$ is some unknown function.
Any and all input is appreciated.