I've not investigated all the particle stuff fully, but it looks like it adds an additional element to the motion of the particles...
A bit like the sinvariance for balls.

Take blackballsRising as an example.

Without the AxialsinOffset the balls just move straight up.
With the AxialsinOffset on axis="x" they also move left and right.

Each value (amp,freq,phaseshift) is given as a pair [min,max]
When each particle is created it gets a random value in that range for that parameter.

HTH

Thanks a lot

But there are also sinanims for particles, aren't there? Like in the rainingLeaves?

Err.. closest thing is these!
There are rotation, movement, scale attributes for the particles, but there's nothing called sinanim in a particleeffect.

Also I think I've been reading it wrong... I'm pretty sure it's Axial Sin Offset
Which kinda tells you everything you need to know.

Oops.

I just checked on my computer with the decrypted files and it is AxialSinOffset.

Other than this and Additive, I think I have particle effects figured out now.