Further, they disagree on who counts as a Jew.
It is hard, for example, for Avigdor Lieberman, the leader of Yisrael Beiteinu, to break bread with the extremist rabbis of Shas and United Torah Judaism, when those rabbis don't regard many of his supporters - immigrants from the former Soviet Union - as real Jews. To them, "Russians" no more belong to the Jewish collective than Palestinians.
Oppressive shadowBut an even bigger obstacle is to be found in the figure of Netanyahu himself, Israel's longest-serving prime minister.
The far-right is largely unperturbed by Netanyahu's trial on multiple corruption charges. Israel's short history is full of major crimes: wars of aggression, forcible population transfer, executions and looting, land theft and settlement building. All Israeli leaders, Netanyahu included, have had a hand in these atrocities. The current focus on allegations against him of fraud and acceptance of bribes looks trivial in comparison.
The far right's problem with Netanyahu is more complex.
He has been presiding over this bloc, relatively unchallenged, since the early 1990s. He has become by far the most skilled, experienced and charismatic politician in Israel. And for that reason, no other far right leader has been able to emerge from under his oppressive shadow.
He may be King Bibi - his nickname - but the far right's more ambitious princes are getting increasingly restless. They are eager to fill his shoes. Their knives are out. Gideon Saar, his Likud protege, created a party, New Hope, to run in last week's election precisely in the hope of ousting his old boss. But equally, Netanyahu is so wily and experienced that he keeps outsmarting his rivals. He has managed to avoid any of his opponent's lethal lunges by exploiting the far right's weaknesses.
Netanyahu has employed a twofold strategy. Despite perceptions abroad, he is actually one of the more moderate figures in the extreme religious and settler bloc. He is closer ideologically to Benny Gantz of Blue and White than he is either to the rabbis who dictate the policies of the religious parties or to the settler extremists - or even to the bulk of his own Likud party.
Netanyahu has become a bogeyman abroad chiefly because he is so adept at harnessing the energy of the religious and settler parties and mobilising it to his own political and personal advantage. Israeli society grows ever more extreme because Netanyahu has for decades provided an aura of respectability, statesmanship and intellectual heft to the rhetoric surrounding the far right's most noxious positions.
In this election he even brokered a deal helping to bring Jewish Power - Israel's most fascistic party - into parliament. If he has to, he will welcome them into the government he hopes to build.
Wearing thinBut Netanyahu's relative moderation - by Israel's standards - means that he has, at least until recently, preferred to include centrists in his coalitions. That has helped to curb the excesses of a purely far right government that might antagonise the Europeans and embarrass Washington. And equally, it has kept the extreme right divided and dependent on him, as he plays its parties off against the centrists.
If the princes of the settlements push him too hard, he can always tempt in a Yair Lapid (Yesh Atid), or a Gantz (Blue and White), or an Ehud Barak (Labor) to replace them.
He has been loyal to no one but himself.
Now that strategy is wearing thin. His corruption trial and the resulting campaign he has waged to weaken Israel's legal and judicial systems to keep himself out of jail has left a sour taste with the centrists. They are now much warier of allying with him.
After last year's election, Gantz only dared join a Netanyahu government after citing exceptional grounds: the urgent need to fight the pandemic in an emergency government. Even so, he destroyed his party in the process. Now, it seems, only a rookie, conservative Islamist leader like Mansour Abbas may be willing to fall for Netanyahu's trickery.
Sensing Netanyahu's weakness and his loss of alternative partners, parts of the far right have grown unruly and fractious.
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