What is the intuition behind Heap's Algorithm?

Question

I am trying to get an intuition for Heap's Algorithm which is used to generate permutations of a given set.

What I can't understand is why if n is even the letter swapped is i and when n is odd the letter shifted is at position 0.

I know it works but what is the intuition behind it ?

How come when n is odd there is always an unique letter in the beginning ? While if n is even the unique letters are from 0 to n-1 ?

The pseudo - code that I am looking at is -:

procedure generate(n : integer, A : array of any):
    if n = 1 then
          output(A)
    else
        for i := 0; i < n - 1; i += 1 do
            generate(n - 1, A)
            if n is even then
                swap(A[i], A[n-1])
            else
                swap(A[0], A[n-1])
            end if
        end for
        generate(n - 1, A)
    end if