You can use OpenSSL and run this command:
openssl.exe s_client -connect www.mysite.com:443
which will use a generic SSL/TLS client to connect to your site and give you a ton of diagnostic info. You can see your entire SSL cert chain, the SSL handshake, SSL session info, etc. Here's a full list of switches you can use with it: http://www.openssl.org/docs/apps/s_client.html
And here's a great online tool to graphically verify your SSL chain is intact: http://www.sslshopper.com/ssl-checker.html

Jason Samuel
Product leader, advisor, and international speaker with 27+ years in enterprise end-user computing, security, and cloud. Has deployed infrastructure at Fortune 500 scale across 34 countries. 1 of 3 people globally to hold Citrix CTP + VMware vExpert + VMware EUC Champion concurrently. 200+ articles, 1,000+ reader discussions.