It's not quite as rare as you might think - it's happened 11 times now.
The most recent instance was in
1997, when England won
the first Test at Edgbaston, but Australia won the series 3-2. The first
time it happened was in England in
1888, when Australia won
the first Test but England, led by WG Grace, won the next two to take
the series 2-1. Australia turned the tables at home in
1897-98, losing the
first Test but winning the remaining four, and they repeated that feat
exactly in
1901-02.
In
1909 England won the
first Test but Australia took the next two, and the five-match series
2-1, but in
1911-12
England turned the tables in a big way, winning the last four Tests
after losing the first one. In
1930 England won the first
Test, but Australia, with Don Bradman in record-breaking form, won two
of the remaining four, while in
1936-37 Australia uniquely came
from 0-2 down to win the five-match rubber 3-2 in Bradman's first series
as captain. In
1954-55, spearheaded by
Frank Tyson, England lost the first Test but won the next three, and in
the next series, in
1956, Australia won the
second Test but lost the series 2-1 as Jim Laker did his stuff. Probably
the most famous occasion was in
1981, when Australia won
the first Test but England, inspired by Ian Botham, took the series 3-1.