What is the name of the part that engages the transmission?

I've been wrenching on cars for over half a century but I'll be darned if I can figure out how to answer your question. Can you be a little more specific about what you're looking for?

First of all, are we talking a manual or automatic transmission? I mean, there's the shift lever, the solenoid pack, the clutch(es), throwout bearing, etc.; all those things could be considered to "engage" the transmission.

What year, engine and transmission is your Jetta?