What about naming the secondary array array_adjoint or adjoint_array to be explicit about its role?

Thank you for this PR @jvdp1. I agree with @jalvesz and like the adjoint_array name.

Regarding your questions:

• I believe the original sort_index is appropriate because it represents an argsort, that is a widely used operation. So I think that the former API would better be maintained (just call sort_adj with [(j,j=1,n)] input for the list). No idea why the former name was chosen, but I think now it would be late to change it.
• In my library, this function is named sort_and_list, I agree it's not a great name. A probably better option could be sort_pair, as in other languages you can do it with a custom comparison function.

These arguments:

could become

         ${t1}$, intent(inout)         :: sorted_array(0:) 
         ${ti}$, intent(inout)         :: adjoint_list(0:) 

to signal that sorting is performed based on the first argument.

Thank you @jalvesz and @perazz for the suggestions.
Based on them I called the new procedure sort_adjoint and the associated array adjoint_array. I kept the name array for the sorted array, as it is used for the other procedures.

sort_index just calls sort_adjoint with an initialised index array.

@perazz @jvdp1 for the test program, maybe something like one of the tests I wrote for the sparse matrices could work?:

    subroutine test_coo2ordered(error)
        !> Error handling
        type(error_type), allocatable, intent(out) :: error
        type(COO_sp_type) :: COO
        integer :: row(12), col(12)
        real    :: data(12)

        row = [1,1,1,2,2,3,2,2,2,3,3,4]
        col = [2,3,4,3,4,4,3,4,5,4,5,5]
        data = 1.0

        call from_ijv(COO,row,col,data)

        call coo2ordered(COO,sort_data=.true.)
        call check(error, COO%nnz < 12 .and. COO%nnz == 9 )
        if (allocated(error)) return

        call check(error, all(COO%data==[1,1,1,2,2,1,2,1,1])  )
        if (allocated(error)) return

        call check(error, all(COO%index(1,:)==[1,1,1,2,2,2,3,3,4])  )
        if (allocated(error)) return

        call check(error, all(COO%index(2,:)==[2,3,4,3,4,5,4,5,5])  )
        if (allocated(error)) return

    end subroutine 

Applying the sorting to row and having col follow. For the sparse matrices I also remove duplicates after sorting, here I think that step is not required.