Okay, so at first I was with you and thought for sure this function was impossible, and I am still not completely convinced this works, but here is what I am thinking.
First the motivation.
f(n)=a for some a
then you know the value of f(a) to satisfy the requirement
but then this sort of cycle of 4 works to fit the definition. I have built these ones to satisfy the requirements. So basically you just gotta find a nice way to partition the integers into dijoint cycles of 4 so this will work out and here is what I came up with.
This is clearly bijective, so it makes sense definition wise.
so basically I think the closed form is
for n positive and odd f(n)=n+1
for n positive and even f(n)=-n+1
for n negative and odd f(n)=n-1
for n negative and even f(n)=-n-1
Okay, I think it works, see what you think.