So I mention something on the invoice like zero rate VAT under a regulation.
If you are using the Reverse Charge (RC) you do not charge VAT on the supply, specifying on its invoice that the RC applies. You account for a notional receipt of VAT at the standard rate as an output and the recover the tax immediately on settlement as an input. You also quote the the other party's VAT Number or equivalent on the invoice.
Otherwise, it's just VAT as usual.
Outside the EU, of course, and that includes Switzerland, the Isle of Man and the Channel Islands, it is an export, a zero rated supply where you still charge VAT, but at a rate of 0%.
Don't blame me, I didn't invent this idiotic system: I've only been accounting for it since it's introduction in the early 70s!