Every time you push the lulav out, the kavana is the same — “I just want You, Hashem.” The intention is to cut through all distractions (gashmiyus and ruchniyus) and connect to the essence of Elokus.
Every time you pull the lulav in, you ask that connection to manifest in a specific way, depending on which of the four sets of nanuim it is. Each of the four sets builds on the previous one (i.e. we need the foundation of humility and ahavas Yisroel to then move on to ask for hislavus and hisorerus, and so on)
Each shaking is done three times in each direction, building intensity with each movement.
The six directions represent our tefillah to absorb that middah in every form and scenario it comes up in (from every “direction”).

This article was found in Nekuda Tova Issue 100 – Sukkos.