TDE installed, LXDE not installed. LXDE launches. TDE will not launch. ?????

Help! I got myself in a mess after updating. The usual, reverting to lightdm and LXDE. Which mess I made worse. Well, I couldn't really see what I was doing. So I purged lightdm. No better. So I purged LXDE. No better. So I purged then installed TDE. aptitude purge tde-trinity aptitude install tde-trinity (The second as per https://wiki.trinitydesktop.org/Debian_Trinity_Repository_Installation_Inst… ) Nothing much happened. Only one package was touched. <aptitude show> says that **LXDE is not installed** and that TDE is installed. It says that lightdm is not installed. It says that tdm-trinity is installed. But when I boot up it is LXDE that launches, bypassing any dm. Yes, I know. That sounds totally mad. But I have repeatedly checked. I am flummoxed. Reporting real error messages is terribly difficult because I haven't got LXDE set up so that I can see it well, and the virtual terminals are impossible. Wiping and reinstalling would be the easiest thing to do, and also the quickest. But it would also be defeatist. Has anyone any other ideas? Thanks, Lisi