I added the columnDefs array to my DataTables initialization code. The purpose is that the column with index 0 can be sorted according to the values in the hidden column with index 1.

After this, the sort order indicators of column 0 no longer work, i.e. they no longer show whether the column is currently sorted in ascending or descending order.
Code:

$('#example').DataTable({
    order: [], // No ordering applied by DataTable during initialisation.
    paging: false,
    columnDefs: [
        { orderData: 1, targets: 0 }, // Order col 0 by col 1.
        {
            target: 1,
            visible: false,
            searchable: false
        }
    ]
});

I suppose the indicators have to be “wired” to column with index 0. But I haven’t found how.