Cricketers have played Tests for England and Australia against each other?

There's only one player who has played both for England against Australia and for Australia against England and it was an awfully long time ago, when the qualification rules were bit more elastic. Billy Midwinter was born in Gloucestershire in 1851, but he was taken to Australia at the age of 9. He played for Australia in the very first Test of all, at Melbourne in March 1877. He was very handy all-rounder but he returned to England later that year, and played for Gloucestershire. When an Australian team toured England in 1878 and he played a few games for them - until in a famous incident he was more or less kidnapped by WG Grace, Gloucestershire's captain, and ‘persuaded’ to turn out for the county. In 1881-82 Billy Midwinter was part of Alfred Shaw's privately raised team which toured Australia, and played in 4 games now considered Tests  before resuming his Australian connections and playing 6 more Tests for them, in 1882-83, 1884 (in England) and 1886-87. Overall Billy Midwinter appeared in 12 Tests, scoring 269 runs at 13.45 and taking 24 wickets at 25.20, with a best return of 5 for 78 in the very first Test of all, at the MCG in 1876-77.

1878 Australian Team and Billy Midwinter sitting in the middle

Billy Midwinter