While the mortgage is important, what supersedes the mortgage is the deed. If your name is the only one on the deed and there are no other liens on the title, then you are the only party with rights to the property. So, since you already said you have a mortgage, your mortgage company has a lien on your property.
Make sense?? If you still aren't clear, let us know and we'll see what we can do to help.
In a traditional sale you are taking title to the property and will be on the deed. This is the paperwork that gets signed at closing. A mortgage really is just a lien on the property that must be paid off. In way the lender does have rights to the property, because if you miss payments they have a right to foreclose (we are mainly a foreclosure by advertisement state) and retake the property as payment.
Very interesting question that I am guessing pertains to a specific issue. This is a very general response, but in reality your title company shall have the best answer that may pertain to a very specific situation you may be in.