i will take a stab at this......
may be slightly not there but here is what i i think,

half turns precede doing quarter turns, so they start them facing forward, do the half turn, do two in plane and she cues the quarter turn with her arms going up, the tricky part is she fades after doing that.

i dont think there is a name for this. but i know the cue of arms up is for the quarter turn and fades work by paying attention and continuing with what the cue suggested.

hope this helps and if i have this wrong pls correct me.
sara

We do a quarter turn with a fade, but not usually right after a half turn. We don't really have a cue for the fade, just the leader stepping back as with other fades.

I didn't watch the clip because I'm at work, but I know what you're talking about. We do it, and yes that is how it's done.
You do the same thing that you do for Turkish Shimmy 1/4 turn, only fade it on the first quarter and the third quarter when you are facing the side walls simultaneously walking the arms up just like the normal TS1/4 turn.

You don't have to do the 1/2 turns first.

We call it the Turkish Shimmy 1/4 turn fade, because well that's what it is!

I don't know if there is a fancier name for it....just Turkish Shimmy ¼ Turns w/Fade. To my knowledge, there is no cue or specified set-up for it. The leader will simply grapevine step with impetus (thanks for the word Megha) toward the back wall to signal a faded version of the Turkish Shimmy ¼ turn. I do teach that, although not required, it’s a really great idea to set up a more complex move with the simpler version of it first. For example, start by doing the Turkish Shimmy ½ turns or ¼ turns before fading, it gets everyone’s brain engaged for possible options. This works particularly well for newer dancers.

In the video clip mentioned, it seemed like the arm cue given just before the ¼ turn fade (@ 2:52) was for the TSWAT but then changed to a ¼ turn fade (possible because she knew the song was about to end). They also do this combo near the beginning of this clip too www.youtube.com/watch and it looks like they use the ½ turns as a set-up move (but again, I don’t think it’s required).

Okay, I'm a little confused about something. I watched the video and I saw the cue for the 1/4 turns as both arms overhead. But I thought this was the cue for a variation of the Turkish Shimmy with half turn and arm sweeps. So what's the difference in the cues for these two variations?