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JavaScript (ES6), 49 45 bytes

Saved 4 bytes with help from @RickHitchcock

f=(n,k=1)=>n<k?[]:[n,k,k+1,n-1,...f(n-2,k+2)]

Demo

f=(n,k=1)=>n<k?[]:[n,k,k+1,n-1,...f(n-2,k+2)]

console.log(JSON.stringify(f(4)))
console.log(JSON.stringify(f(8)))
console.log(JSON.stringify(f(12)))
console.log(JSON.stringify(f(16)))
console.log(JSON.stringify(f(20)))

Non-recursive, 51 bytes

n=>[...Array(n)].map((_,i)=>[2*n-i,,++i][i&2]+1>>1)

Demo

let f =

n=>[...Array(n)].map((_,i)=>[2*n-i,,++i][i&2]+1>>1)

console.log(JSON.stringify(f(4)))
console.log(JSON.stringify(f(8)))
console.log(JSON.stringify(f(12)))
console.log(JSON.stringify(f(16)))
console.log(JSON.stringify(f(20)))

05AB1E, 9 8 7 bytes

L`[Žˆrˆ

Try it online!

Explanation

L           # push range [1 ... input]
 `          # split as separate to stack
  [Ž        # loop until stack is empty
    ˆ       # add top of stack to global list
     r      # reverse stack
      ˆ     # add top of stack to global list
            # implicitly display global list

Python 2, 99 93 88 58 56 55 bytes

f=input()
for i in range(1,f/2,2):print-~f-i,i,i+1,f-i,

Try it online!

_{-6 bytes by removing unneeded indentation, thanks Oliver Ni}

_{-5 bytes by changing the conditional, thanks Luis Mendo}

_{-30 bytes by optimizing the print statements, thanks Arnold Palmer}

_{-2 bytes by putting the loop on one line, thanks nedla2004}

_{-1 byte by doing some wizardry, thanks Mr. Xcoder}