Dean, now we're getting somewhere!
When I set
$sys_cmd = "echo > /root/itworks"
the file is NOT created.
When I take the generated $sys_cmd that was emailed to me in the earlier test, and run it from a telnet connection, it runs okay and the confirming email is delivered to my mailbox. This suggests wget is there and doing its job, and so is bounce.php.
So, it appears that system() in bounce.cgi is not doing its job, though I can't imagine why. I'm running Perl 5.6.1. Any ideas?