They own the IP, so they own the reverse DNS space. It's possible (although very unlikely) that they have delegated it to you. You can find out what DNS server is supposed to have your reverse DNS entries by doing dig -x <ip address>. The AUTHORITY SECTION: will contain the nameservers responsible for your IP.
You own the forward DNS because you own the domain name, but IP addresses aren't bought the same way as domain names. They belong to the provider in the vast majority of cases.
Last edited by chort; 03-01-2009 at 02:22 AM.