The נענועים are an integral part of the mitzva of lulav (see Mesechte Sukkah 37b). But how is that waving supposed to be done? As anyone who’s ever been in a shul over Sukkos knows, the question seems to inspire a wide range of answers. The gemara’s conceptual source is the waving associated with the מלואים (see שמות כט:כז). There, the הנפה and הרמה referenced by the verse suggest to ר’ יוחנן that we should move the four species outwards and inwards, and then up and down.
One popular interpretation involves shaking the lulav with each movement. The source for this seems to be the Rema (תרנא:ט) who wrote:
ומכסכס הלולב בכל נענוע
The popular translation of כסכס is “shake.” However, from the Yerushalmi to Sukkah (פרק ג הלכה ח) this would not seem to be correct:
תני צריך לנענע ג’ פעמים ר’ זעירה בעי הכין חד והכין חד או הכין והכין חד או הכין הכין חד
One must wave (the lulav) three times. Rabbi Zeira asks: this way is one and this way is one, or this way and this way is one? (i.e., are those three waves made up of three sets of “in and out” movements, or one inward movement, one outward, and a third inward).
The Yerushalmi answers via a proof from the laws of Niddah (found in the Bavli in Niddah 62a). There, the required steps for properly cleaning a stain involves כסכוס three times in each direction. Here’s how the Yerushalmi concludes:
תמן תנינן צריך לכסכס ג’ פעמים בין כל דבר ודבר ר’ זעירה בעי הכין חד והכין חד
Which would seem to clearly limit the Rema’s ומכסכס to simple outward/inward/upward/downward movements. And there seems to be no source for shaking.