Mikepier writes:

My neighbors on either side of me have identical Lennox units that say
" turn 1/12 of a turn after making contact with the gasket".
What are these caps for and why is it critical to tighten them
correctly?

Sounds like a king valve. Used to isolate system components when you need
to open up the system. They seal to the atmosphere if all-the-way-open or
all-the-way-closed, while partly-open leaks very slowly; thus the
instructions for opening and closing.