How do I reinstall Unity?

I tested out Unity 5, and of course, it broke things, so I tried downgrading by

sudo apt-get install ppa-purge
sudo ppa-purge ppa:unity-team/staging

It removed Unity completely.

apt-get install unity

seemed to work, but the login screen now offers only Gnome.

How do I get back to Unity 4?

plus one for me. But with this, I got it:

sudo apt-get update
sudo apt-get install --reinstall ubuntu-desktop
sudo apt-get install unity

Seems to wired, and I think it was more luck than brain, but it worked for me.

sudo apt-get update
sudo apt-get install --reinstall ubuntu-desktop
sudo apt-get install --reinstall unity

Try

sudo apt-get update

It looks like you were still running 5.0 and then you just purged it and reinstalled it.

Either way, it did work for me. You might need to Ctrl + Alt + F1 at the login screen and try it through tty1.