What's the best way to rotate objects identically in their own direction? How to rotate these sticks evenly so that they roughly stay in place. I tried to use the array modifier and make them separate objects, and also somehow rotate them in relation to the 3D cursor, but it didn't help.
It seems to work fine:

Select the stick called Cylinder.002. Rotate it as much as you want on the X axis, move it until it's correctly placed:



Delete the other sticks, select 002, press AltD to link-duplicate, in the Transform Pivot Point menu choose 3D Cursor, rotate 120° on the Z axis:



Do it again.

If you need to animate the sticks, first parent the initial stick to an empty, then duplicate both the empty and the stick, move and rotate, and you'll animate each stick: