Dizzy integer enumeration

SOGL V0.12, 8 6 bytes

I».»⌡±

Try it Here! or try the first couple numbers (changed a bit so it'd work)
0-indexed.

Explanation:

I       increment the input
 »      floor divide by 2
  .     push the original input
   »    floor divide by 2
    ⌡   that many times
     ±    negate

Or simpler:

(input + 1) // 2 negated input // 2 times
        I     »     ±      .     »    ⌡

Python 2, 26 24 bytes

lambda x:-~x/2/(1-(x&2))

Try it online!

JavaScript (ES6), 18 bytes

1-indexed.

n=>n/(++n&2||-2)|0

Demo

let f =

n=>n/(++n&2||-2)|0

for(n = 1; n < 20; n++) {
  console.log(n + ' --> ' + f(n))
}