As I understand it, there is also some monkey business related to patents. Something like this: Apple sells its patents to Apple Ireland for a dollar. Then they pay "royalties" to Apple Ireland everytime they sell an iPhone. I can't confirm where I read this.
The most powerful politician in the EU - Jean-Claude Juncker - is personally responsible (not by himself, of course) for setting up these sorts of structures, and making sure they stay in place [1].
A quote from [1]:
"The president of the European commission, Jean-Claude Juncker, spent years in his previous role as Luxembourg’s prime minister secretly blocking EU efforts to tackle tax avoidance by multinational corporations, leaked documents reveal."
So:
> I think a lot of tax agencies simply don't have the muscle to push forward on this
No. They do. They're just prevented from doing so by the "socialists" (hah !) Europe "elected" (just pointing out that the leadership of the EU commission, the only organisation with real law-giving power in the EU aren't directly elected at all).