Polly Toynbee: May has nothing to fear from the rabble across the floor

The person Theresa May trounced most crushingly was on her own side – her predecessor. Serious and commanding, she showed how PMQs should be done – with forensic fact and deadly precision alongside flick-knife jabs. How far this was from David Cameron’s red-faced bluster and bullying refusal to answer any question.

Worse, she confirmed all she had said on the steps of Downing Street about the state of the country she has inherited – socially unjust, hard on working-class boys, with a racially discriminating justice system. That’s a knife between her predecessor’s shoulders too.

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But nothing Theresa May said in her first PMQs was pronounced with such sincerity as her wish for these exchanges with Jeremy Corbyn to continue for a very long time. Of course she hopes he’ll stay. Though she had plentiful fun with Labour spending three weeks to choose a “unity” candidate, with months of infighting still to come.

She had every right to revel in being a second Tory woman PM – bragging she got there through an “all woman shortlist” with no quotas. What does the Conservative party do for women? “It makes us prime minister!” The Labour MPs opposite her could only writhe, with so many more good women on their benches than sitting behind May – but no leader.

Corbyn set out on the Orgreave inquiry – a strong cause, but it instantly cast him back into the 1980s, his natural home decade. He was right to challenge May’s pieties on entering Downing Street – that’s the benchmark she has set herself, and it’s bound to fail because by their nature Tory governments can never favour the poor.

But no, she has nothing to fear (yet) from the rabble across the floor with a leader whose colleagues mostly watched in glowering silence. All her problems sit behind her: Edward Leigh was first out of the traps demanding she commit not to join the single market – before the first breath is drawn in negotiations. Nothing will ever satisfy those Euro-obsessives who have destroyed three of their prime ministers and may well claim the elegant scalp of a fourth.

Ayesha Hazarika: As a Labour supporter, it was excruciating to witness

Theresa May had a brutally brilliant PMQs debut. To be fair, first outings generally go well and she had an embarrassment of riches gifted to her by a Labour party falling apart at the seams. But sometimes PMQ open goals can be easier to flunk than people think.

Yet May rose to the occasion and hit the back of the net again and again. As a Labour supporter, it was excruciating to witness. She was confident, not referring to notes, and exuded cool authority – but most importantly showed she has the capacity to think on her feet and capitalise in an opportunity to land a punch. And Jeremy Corbyn gave her plenty of those. He kept moving between topics and asking open questions, which just allowed her to trot out her top-line positive messages and then go on the attack.

The worst moment for Corbyn was when he asked an earnest question about job insecurity and bad bosses without any self-awareness. Tom Watson’s face showed he knew exactly what was coming. May resisted going for the obvious gag straight away, which made it all the more painful. Her riff about bad bosses who won’t listen to workers and exploit the rules was both funny and politically true, which is why it worked. It was clearly prepared (so hats off to her team) but she delivered it effectively, at the right moment and with a touch of theatrical flourish that she clearly enjoyed and the chamber loved.

May made a smart call to end her exchange with a bigger political message about how while Labour will spend the summer fighting each other, the Tories would crack on with running the country. It was a performance to make Tory MPs feel confident that they picked the right leader and Labour MPs feel the very opposite.

Mark Wallace: May paired jokes with a targeted brutality

This may have been Theresa May’s first PMQs, but she’s no novice at the dispatch box. Six years as home secretary, batting off questions on the most difficult to handle of all government departments, was evidently good preparation for the top job.

The key difference between her previous role and this one is that a lot more people are watching. She betrayed the occasional hint of first-day nerves, but is already well on the way to establishing a PMQs style all of her own. Jeremy Corbyn, as hapless as ever, didn’t stand a chance.

For a politician with a rather dry reputation, May’s deployment of jokes will have surprised some viewers – though unlike Cameron she paired them with a targeted brutality. Tim Farron’s reflection on their time spent fighting North West Durham was met with a stinging crack at the fact that while both are now party leaders: “My party is a little bit bigger than his is.”

The defining moment came when Corbyn asked about unscrupulous employers. In response, she teased Labour MPs that they might recognise the experience of a boss who doesn’t listen to his workers. The answer was remarkable less for its content than its sound – her final words suddenly took on the resonant tone of Britain’s only other female prime minister. It was a spine-tingling moment.

Perhaps she deliberately chose to conjure with the memory that still dominates the modern Tory party, embracing the inevitable comparison on her own terms. Or perhaps it was inadvertent, Margaret Thatcher pressing her way back into the Commons whether May likes it or not. Either way, this was the birth of a new political era – May’s here, and she’s fully in command.

Joseph Harker: We know Corbyn can do better than this

This was a day when not only Theresa May, but her opposite number Jeremy Corbyn, were under the spotlight – given that he now faces the agreed candidate of his own parliamentary party in a leadership contest. May even offered him encouragement: “I hope we’ll be having those exchanges for many years to come,” she said.

Corbyn needed to put in an inspiring performance, to show his MPs and his wider party what they’ll miss if they vote him out. And though May has only been in place for a week, he had plenty of material to go on: her words on the steps of Downing Street before moving into No 10, about fighting the “burning injustices” that face black people, women, working-class kids and those with mental health problems; and the decisions she’s made in the past week, in particular in assembling her new-look cabinet.

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So, for his first question, why did he choose to ask about Orgreave: a serious issue but one that is three decades old, and plays to the notion that Corbyn’s politics are buried in out-of-date trade union disputes? And if it was such a big issue, why didn’t he ask a follow-up question? Instead, Corbyn chose a different topic for each of his six questions: home ownership; Boris Johnson’s racist comments; the long-term economic plan; job security; and child poverty.

On each of these the well-prepared prime minister had either a sharp reply or an evasive response to nullify the attack. And, just as with her predecessor, her strongest moments were in goading her opposite number on his own difficulties: “An unscrupulous boss who doubles his workers workload… remind him of anyone?”

Surely Corbyn has seen enough of politics that he knows humour goes a long way (long enough to take Britain out of Europe, in the case of Johnson). Instead, replying that “I don’t suppose there are too many MPs who go to a food bank” sounds self-righteous, more at home in student politics.

Last week, in David Cameron’s final PMQs, Corbyn played a blinder: full of warmth, humour and self-deprecation. This week that seemed to disappear. We know he can do better than this: if he’s to hold off Owen Smith’s challenge he will need to.

May’s one moment of weakness came in her very last line of the session, replying to Lib Dem leader Tim Farron, who’d warmly recalled their days unsuccessfully contesting a seat in Durham. Entirely out of keeping with the moment, she hit out: “My party’s a little bit bigger than his is!” A tetchy, bullying, mean-spirited line that revealed that below her assured exterior lies a darker, more malicious force. If the public sees more of this, her honeymoon period could soon start to evaporate.