I figured out a solution, since I too am having problems with it. Here's the code which will now sort strings:

use Wx;

package MyFrame;

use strict;
use vars qw(@ISA);

@ISA = qw(Wx::Frame);

use Wx qw(:everything);

my %hash = ();
my $sort = 'asc';
my $acol = 0;


sub new {

    my( $class ) = shift;
    my( $this ) = $class->SUPER::new( undef, -1, $_[0], [ @_[1, 2] ],
                                    [ @_[3, 4] ] );

    $this->{LISTCTRL} = Wx::ListCtrl->new( $this, -1, wxDefaultPositio
+n, wxDefaultSize, wxLC_REPORT );

    $this->{LISTCTRL}->InsertColumn( 0, "Column 1" );
    $this->{LISTCTRL}->InsertColumn( 1, "Column 2" );
    $this->{LISTCTRL}->InsertColumn( 2, "Column 3" );


    # control hidden to speed up insertion
    $this->{LISTCTRL}->Show( 0 );

    my( $id );
    my( $elements ) = 5;

    foreach ( 1 .. $elements - 1 ) {

    # WORKS
    my $nnnn = int(rand(100));
    # my $data = int(rand(100));
    #my $data2 = int(rand(100));
    #my $data3 = int(rand(100));

    # DOES NOT WORK
    my $data = "STRING: " . $nnnn;
    #my $data = "STRING: " . int(rand(100));
    my $data2 = "STRING: " . int(rand(100));
    my $data3 = "STRING: " . int(rand(100));

    $id = $this->{LISTCTRL}->InsertStringItem( $_, $data );
    $this->{LISTCTRL}->SetItemData( $id, $nnnn );
    
    $this->{LISTCTRL}->SetItem( $id, 1, $data2 );
    $this->{LISTCTRL}->SetItem( $id, 2, $data3 );

    $hash{$nnnn}{0} = $nnnn;
    $hash{$nnnn}{1} = $data;
    $hash{$nnnn}{2} = $data2;
    $hash{$nnnn}{3} = $data3;
    
    }

    $this->{LISTCTRL}->Show( 1 );

    my( $item ) = Wx::ListItem->new;

    $this->{LISTCTRL}->SetColumnWidth( 0, wxLIST_AUTOSIZE );
    $this->{LISTCTRL}->SetColumnWidth( 1, wxLIST_AUTOSIZE );
    $this->{LISTCTRL}->SetColumnWidth( 2, wxLIST_AUTOSIZE );

    my( $topsizer ) = Wx::BoxSizer->new( wxVERTICAL );

    $topsizer->Add( $this->{LISTCTRL}, 2, wxGROW );
    $topsizer->Add( $this->{LOGWINDOW}, 1, wxGROW );
    $this->SetSizer( $topsizer );
    $this->SetAutoLayout( 1 );

    use Wx::Event qw( EVT_LIST_COL_CLICK );

    EVT_LIST_COL_CLICK( $this, $this->{LISTCTRL}, \&OnSort );

    $this
}

sub Cmp {

    my( $f, $s ) = @_;

    my( $i1, $i2 ) = ( $hash{$f}{$acol+1}, $hash{$s}{$acol+1} );
    
    print "$f => $s\n";

    if ($sort eq 'asc') {
        if ($i1 lt $i2) {
            1;
        }
    }
    elsif ($sort eq 'desc') {
        if ($i1 gt $i2) {
            1;
        }
    }
}

sub OnSort {

  my( $this, $event ) = @_;

  $acol = $event->GetColumn;

  # use Data::Dumper;
  # print Dumper(%hash);

    if ($sort eq 'asc') {
    $sort = 'desc';
    }
    else {
    $sort = 'asc';
    }

    $this->{LISTCTRL}->SortItems( \&Cmp );
}


package MyApp;

use strict;
use vars qw(@ISA);

@ISA = qw(Wx::App);

sub OnInit {
  my( $this ) = @_;
  my( $frame ) = MyFrame->new( 'wxPerl ListCtrl Test', 50, 50, 450, 34
+0 );

  $frame->Show( 1 );

  $this->SetTopWindow( $frame );

  1;
}

package main;

my( $app ) = MyApp->new;
$app->MainLoop;
[download]

It's a hack, but it works. This ListCtrl stuff has been driving me nuts!!!!!!