This page will allow you to check whether you have a working IPv6 and IPv4 internet connection.
I operate two additional subdomains, ipv4.jamieweb.net and ipv6.jamieweb.net. These are only accessible over their corresponding IP versions, and just host simple pages in order to test your connection.
This page will try to load some CSS from each of the test subdomains. If your browser is able to connect, the style will load and be applied, which will change the text above to "Yes" in green from the default which is "No" in red.
I am looking into setting up a basic API in order to check your connection, the details will be available here once complete.