Lots of good answers on why one would reverse the lathe. But I don't see a good answer for why you would reverse the leadscrew.
The use of the leadscrew reversing lever is to reverse the
relationship between the spindle and the leadscrew. With the spindle turning counter-clockwise(normal forward) the forward drive of the leadscrew cuts from
right to left(toward the chuck), Reversing the leadscrew will cut from
left to right(away from the chuck) That's why my first modification to an Asian 9X19 was a way to reverse the feed. See photos at
http://www.hudsontelcom.com/9X20Gear.html The intent was to gear down the speed of the spindle. But there
are good photos of the reversing mechanism.
There are several reasons for doing this, the most prominant being cutting a left hand thread. Starting the thread at the chuck end with the chuck turning CCW and the threading tool cutting away from the chuck, the thread will come out backwards. Left hand, or reversed.
Knurling is another use,
for me. With the leadscrew running out, there is no way to crash the tooling into the chuck. Saves running the lathe in reverse. I do have a pin to "lock" the screw-on chuck, but this way prevents an accidental loosening. And is much faster set-up time.
Reversing the entire machine is a wholly different issue. Running in reverse means the tooling must be upside down or on the backside of the cross-slide. In my case, my machine is very loose and I cut small threads. 24 TPI is large for me, 40 and smaller is the norm. The easiest way to maintain registration of the thread is to back the tool out and reverse the lathe. Then run the tool back in and set depth of the next cut. Essentially the same as if cutting a metric thread. It's a pain in the wazoo to thread this way but beats rebuilding a 70~ year old machine that I don't have the time,
or money, to do.
I could go on with this answer but won't. It should explain what you are asking well enough.
Bill Hudson
.