For the archives, and for anyone else who makes the same stupid mistake I did.

I have my TV connected computer set to boot up without requiring a password.  
It is running Jessie and TDE14.

I forgot, before I started an upgrade, to set tdm back to requiring a 
password.

O.K.  No-one but me could be so dumb, but still, for the archives:
I ran an update -> upgrade 

 - and forgot to reset the dm to tdm (instead of ldm) before rebooting.  Yes, 
I can be dumb.

When I rebooted after the upgrade, I was in ldm and LXDE.  As far as I could 
see, ldm offered no choice of desktop.  I managed to change back to tdm - but 
tdm wasn't asking me to log in, so wasn't giving me a chance to change to 
TDE.  So there was I, stuck.  In neither tdm nor ldm could I change back to 
TDE.  I could have ldm, a login screen and no choice of session, or tdm and 
no login screen.

Solution?  A solution, anyway.

# aptitude purge tdm-trinity && aptitude install tdm-trinity

After the first half, it closed my xserver down, but gave me a virtual 
terminal login.  So I logged in as root and repeated the command.  (Up arrow 
worked.)

Having found no tdm to purge, the first time round having obviously got that 
far when it closed the xserver down, it installed tdm.

So, set dm to tdm.
Cross fingers.  
# init 6
And there, large as life and twice as lovely is my login screen!  

So, all is well, and I am back in TDE.  I haven't listed everything I tried 
nor everything I called myself.  But hopefully anyone else who gets him or 
herself in as stupid a bind will find this, before going through an hour or 
two of frustration.

Let's just hope that, unlike our institutions, I actually learn the lessons 
that have been learnt!

Lisi