A Letter of Relinquishment (LOR) acts as proof of termination from your publisher. For example, if you are registered with a publisher and then leave that publisher, they will provide you and all other right societies with an LOR to inform all parties you are no longer in an agreement with this publisher, thus they no longer administer your musical works.
A lot of times when you switch from a publisher to Songtrust, rights societies will ask us to provide an LOR from your old publisher before we can begin claiming your works on your behalf. Often times, a Letter of Direction (LOD) is not enough for us to move forward with our claims for your works if another publisher is already claiming them. Thus, without an LOR, your music often goes into conflict when we try to claim it because rights societies don't know if you have terminated your agreement with your old publisher (see our article here on conflicting claims here).
LORs come in many different forms, and can range from a signed document, to a simple email from your publisher stating they relinquish all claims for your works. Whatever the form of your LOR, it is important that you save in the event your works go into conflict between publishers.