A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd.
How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$?
"$\color{red}{\text{Positive/Negative}}$ Reciprocal $\color{blue}{\text{(anti)symmetric}}$"..?
edited according to the comments...