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However, Mr Atkinson believes Fidesz is a worry in itself: "I cringe when I see some in the press refer to Fidesz as 'centre-right.' They're not. They're nationalist populist, what in German is sometimes referred to as a 'Volkisch' right."
Fidesz, itself once a party of youth – the name is an acronym of Fiatal Demokratak Szovetsege, or Alliance of Young Democrats – was founded in 1988 as a libertarian anti-Communist party, joining the Liberal International in 1992. After a poor showing in the 1994 elections, it switched allegiance from liberalism to conservatism.
The British anti-fascist editor says Mr Orban is a political chameleon, shifting ideology to whatever will keep him in power. He notes that the party has been in coalition with Jobbik at the local level in "around 100 municipalities."
In 2008, Fidesz MP Oszkar Molnar, who was also mayor of Edeleny, a town in eastern Hungary, famously claimed that pregnant Roma women take medication to give birth to "fools" to receive higher family subsidies. "I have checked this and it's true; they hit their bellies with a rubber hammer so that they'll give birth to handicapped kids."
Responding to Mr Molnar's statement, Mr Orban said only that his speech was "embarrassing," although the MP was later dropped from national lists and quit the party.
But Mr Kuli, from the Poltical Capital think-tank, thinks the conservative party should be given the benefit of the doubt for now.
"Fidesz has dealt with these issues very gingerly, it is true. But it's a political calculation. They do not want to alienate their rural support by taking decisive action against Molnar and send them into the arms of Jobbik. I don't necessarily support this strategy, but this is why they acted this way," he said.
The Times' Mr LeBor believes that to call Fidesz "Jobbik lite" is "a complete nonsense. During the election campaign it made great efforts to distance itself from Jobbik. Like all parties, they include a wide range of views. Some of their MPs are more right-wing than others."
The Budapest Jewish Youth Organisation's Mr Schonburger also does not think Fidesz are wolves in centre-right sheep's clothing, but he does wish the party was more forthright in countering Jobbik: "We are hoping for the incoming government to make a clear statement on the issue in the next two weeks, Orban needs to make more public statements about what has happened. But I don't know if they are willing to. We certainly don't see any clear statement coming from them yet."
He is however furious at the post-Hungarian election triumphalism of other conservative parties across Europe: "How can they call this a victory? How can they celebrate when so many people have voted for Jobbik?"
"This is the most important issue facing Hungary, maybe even more important than the economic crisis. Something is going on here. How in the middle of Europe, in this new part of the EU, can we have such radical voters, Jobbik, the Hungarian Guard?"
Analyst Mr Kuli is more optimistic: "What will Hungary look like in two years? Orban says he will be able to renegotiate the terms of the IMF deal. But he has very little room to manoeuvre. At the same time, the stability that a supermajority gives Fidesz – it will be the most stable government since 1990 – shouldn't be forgotten. And the economic gurus around him are very intelligent men."
But even he issues a warning: "The trick is if Orban goes further with austerity measures. If his voters see that he doesn't have their dreams in mind, I think you'd see protests against him as well.
"And Jobbik will be ready and waiting in the wings."
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