Hardline Brexiteers are Cavaliers because we have a hardline Brexit government, and they are reverse-engineering their views backwards from that - but it's worth bearing in mind that it didn't have to be this way. The authoritarian and liberal views of the British constitution are much older than Brexit, and it is only chance that has put the Brexiteers on the Cavalier side.
The basic problem
By "Cavalier", I don't mean someone who supports Charles I and the divine right of kings. Today's Cavaliers obviously believe in democracy rather than hereditary monarchy. But the deeper continuity with the historical Cavaliers is a belief in strongman leadership and suspicion of Parliament as an independent actor. In the Cavalier worldview, Parliament is there to support the government. MPs' job is to boo and cheer at each other across the chamber while ministers get on with the serious business of running the country. The soundbite used to describe this is: "Parliament must have its say, but the government must get its way".
"Roundheads" regard this idea as a constitutional obscenity. They are supporters of parliamentary and constitutional governance, complete with checks and balances. They believe that the executive should be constrained by a powerful legislature and by independent judges. In the context of Brexit, they may well believe that Brexit needs to be delivered because the referendum created a moral obligation to do so, but they do not believe that the referendum result justifies violating orthodox constitutional principles or the rule of law.
As I say, this divide has existed for a long time, but it has tended to be masked because in the British political system the executive is generally aligned with the legislature. It is unusual for the party that controls the executive not to be able to command a majority in the House of Commons. Between 1945 and 2017, this has been the case only on three brief occasions: during Harold Wilson's brief third government in 1974, and in the dying years of the Callaghan and Major administrations.
The fact that the executive and the legislature are generally controlled by the same party means that Cavaliers and Roundheads can generally both appeal to Parliament as the ultimate political authority. This is often linked with romantic notions of Parliament as the historical source of English law and liberties, disappearing into the mists of Magna Carta and the middle ages. The problem is that the two sides have meant different things by "Parliament". Cavaliers could afford to idealise Parliament when Parliament was controlled by the executive. Roundheads thought that Parliament ought to count for something in its own right.
It is this divide which has been exposed by Brexit.
It is a divide that exists in other countries too. Some other European states - France being one prominent example - have a rightwing anti-parliamentary tradition that goes back decades. Of course, everybody in the modern West believes in democracy (or says that they do), so the divide isn't between democracy and monarchy or democracy and dictatorship. Even Vladimir Putin holds elections, and Russia experts say that he would probably still win them even if they were conducted fairly. The same can be said of Orban, Kaczynski, Erdogan, Bolsanaro and the rest. The divide is rather between an authoritarian form of democracy, in which the only mechanism of accountability is when the strongman and his regime face a five-yearly general election - and a liberal form of democracy, in which the government has to worry not only about losing a distant election but also about losing a vote in the parliament or a case in the supreme court today.
The problem with the Cavalier view
The fallacy at the heart of the Cavalier view is that we elect governments. To believe this is to make a fundamental mistake about the British constitution. In some countries - those with a presidential system of government - voters do indeed elect the executive. This is the case in America, France, South Korea, Argentina and many other countries. But in Britain, we elect MPs, and MPs choose the government. The Commons is the "master of the ministry", to use the historic phrase, and this is why it is perfectly acceptable for the government to change without an election (Blair/Brown, Cameron/May and May/Johnson being the most recent examples of this).
The result of this is that only MPs have direct democratic legitimacy. The democratic legitimacy of the government is secondary and indirect. If there is a conflict between the government and the House of Commons, our constitutional tradition requires that the government must lose that conflict. And that is true however much you want Brexit.
The notion that we do not elect governments is obscured in practice by the fact that MPs are almost invariably party members, and the two main party leaderships campaign on the basis that they will form a government if the election results favour them. But these are matters of political practice. They have no bearing on the constitutional principles involved. Parties have no constitutional status, as shown by the fact that MPs can and do defect from one party to another without having to fight a by-election. Historically, governments have changed between different parties and coalitions without an election being needed (there are several precedents for this from within the last century).
Political parties are a necessity - a necessary evil, we might say - in an advanced democracy. But emphasising their role at the expense of the independence of individual MPs is essentially an authoritarian, pro-executive move. Strengthening the role of parties means, in practice, strengthening the powers of patronage and discipline wielded by the party leaders. Another variant of the same move is emphasising the authority of party manifestos: manifestos are written by the party leaderships and implemented by the executive. Hardly any voters read manifestos, and they have no legal effect. MPs are absolutely free to rebel against them. The only convention, let alone rule, which applies to party manifestos is the Salisbury convention, which applies to the House of Lords rather than the Commons (it holds that the Lords cannot block a party's manifesto commitments).
Of course, the reason why MPs don't normally rebel against their party or manifesto is down to the power of the whips. Traditionally, when Parliament was run as a kind of boys' club, whipping was not necessarily a problem because the relationship between the leadership and backbenchers worked both ways. MPs did not need to rebel in the division lobbies as long as they could influence the government privately. Ken Clarke wrote in his memoirs, Kind of Blue:
The government was profoundly influenced by the opinion of its backbenchers and expected to make concessions on the content of bills and on policy positions if any significant disquiet was detected. A rebellious vote was very rare indeed.... Concession and compromise out of the public gaze in response to influential Members' opinions was quite normal.Let that last sentence sink in.
The result of this gentlemen's club-like atmosphere was that Parliament was profoundly more influential on the government than it is today. I spent my career observing the steady decline of Parliament's influence on public life....
The Benn Act
Brexit has lead to some wild constitutional theorising. We are told that it was illegitimate for the Commons to pass the Cooper Act, and then the Benn Act, as neither measure was proposed by the government. But the notion that only governments can or should propose legislation is a complete non-starter.
The clue is in the word "legislature". The legislature is there to legislate. But on the Cavalier view, it is essentially a talking shop. It serves to "hold the government to account" - not by actually telling the government that it cannot do things that it wants to do, but by... asking questions to ministers. Much like journalists do at a press conference.
Any objections to the Cooper and Benn Acts fail in the light of the established fact that private members' bills are a recognised part of the Parliamentary process. They are regularly proposed and not infrequently enacted. Eight were passed in the last (2016-17) Parliamentary session. In most cases, such bills are uncontroversial; but not always. It was private members' bills that abolished hanging, legalised abortion and decriminalised homosexuality (against the will of the majority of voters at the time, it might be added).
Cavaliers answer this by saying that private members' bills are effectively government bills because in practice they are only passed if the government allows parliamentary time for them to be debated. But this is insufficient to meet the Cavaliers' own case. Private members' bills appear in no manifesto, and the executive cannot be held accountable for them. They may also be actively displeasing to ministers, as demonstrated by Tony Blair's widely reported unhappiness at the passage of the Hunting Act 2004.
Where to now?
The best-case scenario at this point is that the Cavaliers have gone too far. A combination of Boris Johnson's boorish arrogance and John Bercow's willingness to stand up for the Commons may lead to a fundamental shift away from the Cavalier model in the future - whatever happens with Brexit. The worst-case scenario is... well, we know what it is.
As the arch-Brexiteer conservative Peter Hitchens once said, "when Parliament comes into conflict with the executive, you don't need to ask me which side I'm on".