Can you invoice in any currency?
Yes. Article 230 of the VAT Directive is explicit: "The amounts which appear on the invoice may be expressed in any currency, provided that the amount of VAT payable or to be adjusted is expressed in the national currency of the Member State, using the conversion rate mechanism provided for in Article 91."
This means you are free to invoice your German client in US dollars, your French client in British pounds, or your Spanish client in Swiss francs. The invoice total, the taxable amount, the line items — all of these can be in whatever currency you and your client agree upon.
There is no EU requirement to display amounts in euro (unless euro is your national currency). There is no requirement to show dual currency amounts. The only constraint is on the VAT amount itself.