Really? There’s no chance of a British Lieberman, then? (Joe Lieberman was, for a while, the most Republican Democrat of them all. If a vote went straight party line, you could count on Lieberman voting with the Republicans even though he was elected as a Democrat and officially counted towards the Democrats’ count of seats.)
If it works like that, why even have people in the seats?
There’s no chance of a British Lieberman. If he was elected as Labour, but often voted with the Conservative Party, he would be expelled from the Labour Party, and have to run against a different person as the Labour Party candidate in his constituency.
And why have people in the seats? Because they are working for their constituents, but in a different – generally more private – way than members of the U.S. Congress. That will include being part of debates within their party. Just because a parliamentary political party looks monolithic does not mean that there are no divisions within the party. And that internal debate can go all the way to replacing the leader of the majority party in the lower house in between general elections – as, for example, happened with Kevin Rudd as Prime Minister of Australia last year. He was dumped by MPs from his own party.