I found a Privoxy bug!
And with the help of a few people, worked around it!
Using Privoxy, I couldn't access a secure (https) site that had a port other than 443 (the default secure port) specified. Nothing would show up in the log, so I couldn't diagnose what was happening.
A person suggested adding this line to my filters:
+limit-connect{443,19638}
It worked! So now the bug is that without that line, an entry should have been posted to the log saying that a secure connection was refused.
I'll leave the fix up to the people who actually understand that stuff.