OK so... wikipedia has a pretty good summary.
You're right. I was wrong. There is no IP licensing payment from the US entity to Ireland.
IP licensing happens between 3rd party countries and Ireland. Irish tax law (to our great pride) gives IP licensing revenue tax exemption. It doesn't count as revenue for tax purposes. Once here, it can be transferred to a proper tax haven like Bermuda. Since its tax free, it doesn't matter that Ireland (like everyone) doesn't recognise the transaction to Bermuda as a legitimate expense. It didn't count as revenue anyway.
I guess that invoicing to Ireland is neither here nor there, just more convenient when the money needs to come here anyway.
Once the cash is in Bermuda, the game is done. The Bermuda company can hold it, buy shares, etc. This is why Apple (And MSFT) have moved all their IP to Ireland though.
https://en.wikipedia.org/wiki/Double_Irish_arrangement
Here's the good bit:
Without such IP, if Microsoft charged a German end-customer, say $100, for Microsoft Office, a profit of circa $95 (as the cost to Microsoft for copies of Microsoft Office is small) would be realised in Germany, and German tax payable. With such IP, Microsoft can additionally charge Microsoft Germany $95 in IP royalty payments on each copy of Microsoft Office, ensuring that its German profits are zero. The $95 is paid to the location in which the IP is legally housed. Microsoft would prefer to house this IP in a tax haven; however, higher-tax locations like Germany do not sign full tax treaties with tax havens, and would not accept the IP charged from a tax haven as deductible against German taxation. The Double Irish fixes this problem.[8][9]
The Double Irish enables the IP to be charged-out from Ireland, which has a large global network of full bilateral tax treaties.[g] The Double Irish enables the hypothetical $95, which was sent from Germany to Ireland, to be sent-on to a tax haven like Bermuda, without incurring any Irish taxation.