The grooved Avon supplied on the RE's looks just like the grooved Avon's that are shown on the front of most of the old BSA's, Triumph's, Velocettes, Matchless's and other British bikes photos from the 1950's. They are a classic design.
Of course, back in the 1950's, 60 mph was the speed limit on the high speed turnpikes in the USA and was probably way over any legal speeds in the UK so maybe the grip wasn't as important as it is now.