I am attaching an example that supports sorting of ListView Columns by implementing the IComparer Interface for the Sort method.
When the column is clicked once it will be sorted in ascending Order. If it is clicked again it will sorted in descending order etc.

Form1 code:

      class-id testlistsort.Form1 is partial
                inherits type System.Windows.Forms.Form.
      working-storage section.
      01 lvwColumnSorter type testlistsort.ListViewColumnSorter.
      method-id NEW.
      procedure division.
          invoke self::InitializeComponent
          *> Create an instance of a ListView column sorter and assign it
          *> to the ListView control.
          set lvwColumnSorter to new testlistsort.ListViewColumnSorter
          set listView1::ListViewItemSorter to lvwColumnSorter
          goback.
      end method.
      method-id Form1_Load final private.
      01 row type ListViewItem.
      procedure division using by value sender as object e as type System.EventArgs.
          set row to new ListViewItem("John Smith")
          invoke row::SubItems::Add("ABC Software")
          invoke row::SubItems::Add("800-555-1234")
          invoke listView1::Items::Add(row)
          set row to new ListViewItem("Ann Johnson")
          invoke row::SubItems::Add("Z-Systems Data")
          invoke row::SubItems::Add("111-222-3333")
          invoke listView1::Items::Add(row)
          set row to new ListViewItem("Zachary Jones")
          invoke row::SubItems::Add("New Life Systems")
          invoke row::SubItems::Add("555-987-1111")
          invoke listView1::Items::Add(row)
       *> sets second row in listview to be selected row as indexes are 0 based.
          set listView1::Items[1]::Selected to true
          goback
      end method.
      method-id listView1_ColumnClick final private.
      procedure division using by value sender as object e as type System.Windows.Forms.ColumnClickEventArgs.
       *> Determine if clicked column is already the column that is being sorted.
          if e::Column = lvwColumnSorter::SortColumn
       *> Reverse the current sort direction for this column.
              if lvwColumnSorter::Order = type SortOrder::Ascending
                  set lvwColumnSorter::Order to type SortOrder::Descending
              else
                  set lvwColumnSorter::Order to type SortOrder::Ascending
             end-if
          else
       *> Set the column number that is to be sorted; default to ascending.
             set lvwColumnSorter::SortColumn to e::Column
             set lvwColumnSorter::Order to type SortOrder::Ascending
          end-if
       *> Perform the sort with these new sort options.
          invoke listView1::Sort
          goback
      end method.
      method-id button1_Click final private.
      procedure division using by value sender as object e as type System.EventArgs.
          invoke self::Close
      end method.
      end class.

ListViewColumSorter Class:
     *>
      *> This class is an implementation of the 'IComparer' interface.
      *>
     $set ilusing"System.Collections"
     $set ilusing"System.Windows.Forms"
      class-id testlistsort.ListViewColumnSorter implements type IComparer.
      working-storage section.
      01 ColumnToSort   binary-long.
      01 OrderOfSort type SortOrder.
      01 ObjectCompare type CaseInsensitiveComparer.
      property-id SortColumn binary-long.  
         getter.
            set property-value to ColumnToSort
        setter.
            set ColumnToSort to property-value
      end property.
      property-id Order type SortOrder.  
         getter.
            set property-value to OrderOfSort
        setter.
            set OrderOfSort to property-value
      end property.
      method-id New.
      local-storage section.
      procedure division.
         *> Initialize the column to '0'
             set ColumnToSort to 0
         *> Initialize the sort order to 'none'
             set OrderOfSort to type SortOrder::None
         *> Initialize the CaseInsensitiveComparer object
             set ObjectCompare to new CaseInsensitiveComparer
             goback.
      end method.
      *> Automatically inserted Methods from System.Collections.IComparer
      *> This method is inherited from the IComparer interface.  It compares the two objects passed using a case insensitive comparison.
      method-id Compare.
      01 compareResult  binary-long.
      01 listViewX     type ListViewItem.
      01 listViewY     type ListViewItem.
      procedure division using by value #x as object #y as object returning return-item as binary-long.
       *> Cast the objects to be compared to ListViewItem objects
            set listViewX to #x as type ListViewItem
            set listviewY to #y as type ListViewItem
       *> Compare the two items
            set compareResult to ObjectCompare::Compare(listviewX::SubItems[ColumnToSort]::Text,
                 listviewY::SubItems[ColumnToSort]::Text)
       *> Calculate correct return value based on object comparison
            if OrderOfSort = type SortOrder::Ascending
       *> Ascending sort is selected, return normal result of compare operation
               set return-item to compareResult
          else
               if OrderOfSort = type SortOrder::Descending
       *> Descending sort is selected, return negative result of compare operation
                  compute return-item = compareResult * -1
               else
       *> Return '0' to indicate they are equal
                   set return-item to 0
               end-if
          end-if
          goback
      end method.
      end class.

Works....of course.

Works....of course.