The gamma function is essentially given by its functional equation $$\Gamma(z+1)=z\Gamma(z),$$ together with $\Gamma(1)=1$. This is a translation formula.
Likewise, there is a reflection formula: $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{z \sin(\pi z)};$$ $$\Gamma(1-z) = \frac{\pi}{z \sin(\pi z)}\cdot \frac{1}{\Gamma(z)}$$
It occurred to me, mostly in jest, that of the four Euclidean isometries (translation, reflection, rotation, and glide reflection), there are known functional equations of $\Gamma$ for each, except rotations, glide reflections being compositions of translations and reflections. So, I am wondering what is known about the function $F_z(\theta)$ given by
$$\Gamma(e^{i\theta}z) = F_z(\theta) \Gamma(z),\qquad 0\le\theta<2\pi$$
Clearly we have $F_z(0)=1$, and from the reflection formula one can deduce $$F_z(\pi) = \frac{-\pi}{z \sin(\pi z)}\cdot \frac{1}{\Gamma(z)^2}$$
The best result I know uses the log-gamma series: $$\log(\Gamma(z)) = -\gamma z - \log(z) - \sum_{k=1}^{\infty} \frac{z}{k} - \log\left(1+\frac{z}{k}\right);$$ for fixed $z\notin -\mathbb{N}$, the series converges since the summand is $O(k^{-2})$. Then we have: $$\log(\Gamma(e^{i\theta}z)) = -\gamma e^{i\theta}z- \log(e^{i\theta}z) - \sum_{k=1}^{\infty} \frac{e^{i\theta}z}{k} - \log\left(1+\frac{e^{i\theta}z}{k}\right);$$ $$F_z(\theta) = \exp\left(\log(\Gamma(e^{i\theta}z))-\log(\Gamma(z))\right)$$ However, I haven't been able to find anything profitable by analyzing this series, even using the Taylor expansion of $\log$.
There is a result for the magnitude of $F_z(\theta)$. For $z=a+bi$ we have $$|\Gamma(z)|^2=|\Gamma(a+bi)|^2 = |\Gamma(a)|^2 \prod _{k=0}^{\infty} \frac{(a+k)^2}{(a+k)^2+b^2}$$Multiplying $(\cos(\theta)+i\sin(\theta))(a+bi)$ and substituting real and imaginary parts gives an equation for $|\Gamma(e^{i\theta}z)|^2$, but it is not super nice.
For special values, the real/imaginary parts show even/odd symmetry, respectively. I will upload pictures later, but it seems that the real part is even about $\pi-\arg(z)$; I haven't found a similar relation for the imaginary part.