My question relates specifically to the UK.

Is it legal for a private company, say a train or bus company, to detain you if they determine that you don't have a valid ticket? Furthermore, if it is legal, what rights do you have in such circumstances? Obviously, if you were arrested by the police you would have a given set of rights, but train companies, and the private companies that run the train stations in the UK, whilst government owned, are private companies (if I understand correctly).