I said in Re^3: HTTPS connection with LWP and self-signed certificate ( openssl/ssldump )

Hmm... https://metacpan.org/pod/IO::Socket::SSL#Common-Problems-with-SSL says

Also, util/analyze-ssl.pl in the distribution might be a helpful tool when debugging SSL problems, as do the openssl command line tool and a check with a different SSL implementation (e.g. a web browser).

So I'd try that next :)