Performance recommendations stuck on 'Validating'
I found in the documentation that sometimes Azure will take 72 hours to validate a performance recommendation. And indeed after 72 hours it finished evaluating and reverted the recommendations.

(It turned out that identifying the indexes that would really make a big difference was quite easy and I didn't need Azure Performance Recommendations. I copied one of the long-running queries into SSMS, chose a representative set of inputs, and then ran Query > Display Estimated Execution Plan. The result showed me one step in the query was taking 99.9% of the time and it even suggested an index to add. Result was the query time came down from 20 seconds to 0.1 second. I realise this will not be news to DB admins but I hope it will help someone.)