Much as pets and their owners develop uncanny likenesses, Theresa May’s Brexit deal conveys the prime minister’s character as well as her politics. It is narrow and brittle, all circumscription and conditional permission. If it were a book, the cover would show a door being closed with care; not slammed but still closed.

No wonder the Tory hardliners hate this vision. They wanted flags fluttering in blue skies, and explain the gloomier picture as a consequence of letting a non-believer run the negotiations. The ultras of Jacob Rees-Mogg’s European Research Group believe that May’s conversion to the leave cause was incomplete. Many pro-Europeans say the opposite – that the deal is a dud because May was too zealous, positioning herself behind the leavers’ sacred red lines. Neither of those views is quite right.

Any Brexit model would be flawed, but May’s deal has bespoke horrors, tailored to the prime minister’s hybrid taste: she is an economic remainer and a cultural leaver. She accepts the Treasury opinion that alignment with European markets is essential for stability and prosperity. But her gut affinity is with voters who see Brexit as a way to control immigration. There is an irresolvable contradiction between the economics of openness to EU trade and the politics of clamping down on EU people, which is why May’s deal is a tortured mess, as is the government that produced it.

Resentment of migrants is not the only cause of Brexit, as those who had other motives for voting leave are eager to point out. But no one who worked on the ground during the referendum campaign doubts that perceptions of an open door to foreigners gave the winning side its momentum in the final straight. “Sovereignty” was the fizzy tonic in the glass; Nigel Farage stoking fear of a swarthy multitude was the gin.

Ministers and officials who have worked with May over many years insist she is not a nostalgic nationalist in the Ukip mould. (She did, after all, vote remain.) Still, her Conservatism has a parochial, illiberal quality that was radicalised during her tenure at the Home Office, partly by the grind of always dealing with stress in the immigration system, partly in reaction against the metropolitan hauteur that she felt in cabinet as disrespect from David Cameron and George Osborne.

Pledges of immigration control have a poor record of dissipating anger. British politicians have been trying it for a generation

May interpreted the leave vote as a demand for cultural redress by people who felt their identity, sense of belonging and security had been trampled in a decades-long march of globalisation. In her now infamous “citizens of nowhere” speech to the 2016 Tory conference, the prime minister described the referendum as a “quiet revolution” and warned that attempts to thwart it would provoke a dangerous backlash. That translated into determination to quit the EU single market regardless of economic consequences, because free movement of labour would be a non-negotiable part of a softer package.

Sure enough, two years later, immigration is May’s top line when selling the deal. She boasts that free movement is over “once and for all”. On Monday she told an audience of business leaders that EU nationals would no longer be able to “jump the queue” for jobs. She did not add that a truncation of European citizens’ rights in Britain was also a limitation of British citizens’ rights in Europe. Hardly anyone does. The idea of free movement as a reciprocal benefit, the ending of which diminishes the value of a UK passport, has not been part of the debate. (It is recognised well enough by tens of thousands of Britons of Irish parentage or other available nationality who have applied for new documents to preserve the privileges of EU citizenship.)

Many Labour MPs share May’s view that Brexit only really means Brexit if it has a tough borders clause. That is also a calculation for opposition strategists who see the electoral road to Downing Street passing through leave-voting areas, where free movement is toxic. Between 2010 and 2015, Ed Miliband flailed around trying to satisfy incompatible demands from, on one hand, an activist base who treated talk of control as the thin end of a racist wedge and, on the other hand, voters who blamed Labour for letting too many people into Britain and didn’t like being accused of bigotry when they raised the matter. Tagging along with Brexit hasn’t resolved that dilemma, but it has drawn some of the sting. It gives Jeremy Corbyn a “tough” migration policy by default, without him having to sell it as such.

Yet free movement does not, in truth, demand zero regulation of labour flows. The UK opted for the loosest application of the policy. Other EU states are fussier, requiring registration, health insurance and proven job offers as conditions of settlement. A report published on Tuesday by Global Future, a liberal thinktank, sets out ways that migration might be managed without ripping the UK out of the shared European economic space from which it has unquestionably benefited. Such technical solutions alone cannot dispel public anxiety around demographic change and cultural dislocation, but they are necessary tools in the debating arsenal. Likewise, abstract assertions that migration is economically advantageous get little purchase with people who suspect the advantages flow out of their communities. The liberal argument has to be couched within a bigger offer of prosperity and opportunity more fairly distributed.

But pledges of immigration control also have a poor record of dissipating anger. British politicians have been trying it for a generation without success. Maybe ending free movement is the big one, the anti-immigration nuke that settles the matter “once and for all”. I’m guessing not. Outside the EU, Britain will still import workers. It will still have first-, second-, third-generation migrant communities. There will still be cynical politicians on the scout for scapegoats once the economic fallout of the Brexit they peddled is felt.

Not every Eurosceptic is obsessed with immigration, but May is. Not every plan has to be inscribed on the wrong side of a closed door, but May’s does. There must come a point when the immigration argument is reopened, when the terms are redefined and the myths refuted. Some seem to think that time can only come after the Brexit bomb has dropped. I rather think it is now.