Jelly, 9 bytes

ŒRṗAS$Þ⁸ị

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1-indexed.

Explanation:

ŒRṗAS$Þ⁸ị Arguments: n, k
ŒR        Return sorted range from -n to n, inclusive
  ṗ       Cartesian power of the range above and k
   AS$Þ   Sort by sum of absolute values
       ⁸ị Get the nth item

Inverse:

AṀ×LŒRṗLAS$Þi

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Explanation:

AṀ×LŒRṗLAS$Þi Arguments: z
A             Get absolute values of z's items
 Ṁ            Max
  ×L          Multiply with z's length (let's call it x)
    ŒR        Get sorted range from -x to x, inclusive
      ṗL      Cartesian power of the range above with z's length
        AS$Þ  Sort by sum of absolute values
            i Return the index of z in the list we made above

Jelly, 15 bytes

µ<1ạḤðŒRṗAS$Þ⁸ị

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1-indexed.

Inverse:

AṀ×LŒRṗLAS$Þiµ:2NḂ}¡

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Jelly, 9 bytes

ŒRṗAS$Þ⁸ị

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Inverse:

AṀ×LŒRṗLAS$Þi

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Jelly, 9 bytes

ŒRṗAS$Þ⁸ị

Try it online!

1-indexed.

Explanation:

ŒRṗAS$Þ⁸ị Arguments: n, k
ŒR        Return sorted range from -n to n, inclusive
  ṗ       Cartesian power of the range above and k
   AS$Þ   Sort by sum of absolute values
       ⁸ị Get the nth item

Inverse:

AṀ×LŒRṗLAS$Þi

Try it online!

Explanation:

AṀ×LŒRṗLAS$Þi Arguments: z
A             Get absolute values of z's items
 Ṁ            Max
  ×L          Multiply with z's length (let's call it x)
    ŒR        Get sorted range from -x to x, inclusive
      ṗL      Cartesian power of the range above with z's length
        AS$Þ  Sort by sum of absolute values
            i Return the index of z in the list we made above