Thanks for all of the helpful replies, everyone. I was using the constants to give names to magic numbers I was passing in query string parameters. E.g.:

something?action=1

...
<%once>
use constant LOGIN  => 1;
use constant LOGOUT => 2;
</%once>

<%init>
if ($action eq LOGIN) {
...
[download]

I think I'll replace the numbers passed in the parameter values with their constant-name equivalent. E.g.:

something?action=login

instead.