You can, after a fashion. It was much harder on Xbox 360. With Xbox One you can load a game to an external hard drive. It's still going to have the DRM and such, but at least it's somewhere.
With the original Xbox Live Arcade circa 2004, I had purchased and downloaded a game called Alien Sky. It's not available to play on Xbox 360 or Xbox One, and I therefore haven't been able to play it. Not only that, but the original Xbox Live servers are no longer active and I'm unsure if I'd be able to reclaim it if desired. Fortunately, pirates have taken care of this situation for me. Still, I shouldn't be able to lose the things that I pay for due to the actions of the company through which I purchased it.
Sorry to hear about your songs. That's lame.