Q: What kind of projection is this?
I found the following maps by investigating a function $\phi(x)=\exp\big(\frac{1}{\log(x)}\big)$ and intersecting it with rays from the origin. The maps essentially swap the two branches of the graph of $y=\phi(x)$.
$$ (x,\phi(x)) \mapsto \big(\frac{1}{\phi(x)},\frac{1}{x}\big),~~x\in(e,\infty) $$
$$ (x,x) \mapsto \big(\frac{1}{x},\frac{1}{x}\big)~~ x=e $$
$$ (x,\phi(x)) \mapsto \big(\frac{1}{\phi(x)},\frac{1}{x}\big), ~~x\in(-\infty,e) $$
By letting rays extend from the origin and intersect $\phi(x)$ the projection from the function to itself becomes clear. Feel free to verify my claim of the above mappings. Here's a picture of the function $y=\phi(x)$ and a visual aid of what's going on:
