$\frac{\left(x-1\right)!+1}{x}=\frac{\left(y-1\right)!+1}{y}$
When I graphed it, I noticed that it bears a resemblance with the equation (which could of course be completely coincidental):
$\frac{x^2+y^2}{x^2y^2}=x^3+y^3+a\left(x+y\right)+2b$ for a=1 and b=-1.
I derived the last expression from messing around with the plane form of an elliptic curve, ($\frac{1}{y^2}=x^3+ax+b$) and then adding together the inverse function of that equation in order to mirror the symmetry of the gamma functions.
Of course I am not talking about the trivial solutions when y=x but rather the less predictable ones, when 0<x<1. (Might have some relationship with the reflection formula?)