I have always considered myself British, but also Anglo-Irish at the same time, I think it is important to remember your roots (but not in the same way as the 8th generation American-Irish do).
My mum came from Ireland but I can say with a degree of certainty that not all of her family were originally from there, her mothers side of the family hailed from Glasgow. As for my grandad, he spun a yarn about being descended from the Viking settlers who lived in Dublin, think he may have...