Reverse Deltas of an Array

Jelly, 6 bytes

I;ḢṚ+\

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How it works

I;ḢṚ+\  Main link. Argument: A (array)

I       Increments; compute the deltas of A.
  Ḣ     Head; yield the first element of A.
 ;      Concatenate the results to both sides.
   Ṛ    Reverse the resulting array.
    +\  Compute the cumulative sum of the reversed array.

Jelly, 5 bytes

.ịS_Ṛ

This uses the algorithm from Glen O's Julia answer.

Try it online!

How it works

.ịS_Ṛ  Main link. Argument: A (array)

.ị     At-index 0.5; retrieve the values at the nearest indices (0 and 1). Since
       indexing is 1-based and modular, this gives the last and first element.
  S    Compute their sum.
    Ṛ  Yield A, reversed.
   _   Subtract the result to the right from the result to the left.

Julia, 24 bytes

!x=x[end]+x[]-reverse(x)

This is the "clever" way to solve the problem. The negative reverse of the array has the "deltas" reversed, and then you just need to fix the fact that it starts/ends at the wrong places.