Liverpool were away 6 in a row (04-05-09/10).
it seems bizarre though. In 02/03 our first 3 home games just coincidentally happened to be against 3 of the top 4. And we started away to Liverpool and home to United back to back seasons. Since then, we've had relatively comfortably starts against midtable/newly promoted teams.
I can't remember the year, but i think at one point we played Wimbledon 4-5 times in a row. Lots of draws as the teams knew each other so well. It was 2 legged League cup, fa cup and one or both league games if i recall. Don't quote me on that as it was long time ago



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