Kim Iversen is very thoughtful IMO. she concluded that the Kurds are similar to Zionist Jews in their hope for some utopia. She said their utopia has a name and is very communal, even communist.
Since most of their presence is in nations created somewhat arbitrarily and the world peed on their promise of 1920 to give them a nation (because they all caved to Ataturk)...it's been a sh*tshow since. Half live in "Turkey" and are poorly treated, heritage denied and erased from history. Same in Syria. In Iran they declared independence and were repressed by the Soviets, who then handed them to the Iranians. Nationalist resurgence resumed in Northern Iraq and Iraq played nice with them until driving them out and planting Arabs instead...and it lead to insurgencies in Iraq and Iran...Iran/Iraq deal sees first US betrayal of the Kurds...Ayatollah slaughters the Iranian Kurds who find an odd ally in Saddam Hussein while the Iraq Kurds rebel against him and he retaliates with chemical weapons, succeeding campaigns see them flee to Turkey & Iran and not until the Northern No-Fly Zone did they regain a presence in Iraq...lots of give and take until The Kenyan's Arab Spring introduces chaos on Arab end leading to Kurds to press their cases everywhere...rise of DAESH (thanks Kenyan prick!) and Kurds have an ally in the US again...everybody in region has allies/enemies in the ISIS years...next thing you know its all a big mess...mostly Turks vs Kurds and everybody against ISIS...Iran starts whacking Kurds in Iraq, Turkey starts whacking Kurds in Syria and Iraq and for the most part the activity remains most prominent in the Iraq/Turkey area and they remain Pro-US despite being cut to the wind...
As for politics, PJAK in Iranian is socialist-named but nobody speaks much about how it is organized, seems like a club. KDP in Iraq seems more "democratic" and has good relations in the West and with Israel. PUK in Iraq is the sum of many socialist groups, and they support self-determination and democracy and ally with KDP on big things and had good relations with the West and Iran...but they folded when Taliabani left and formed the Gorran Movement (liberal reformist movement fighting corruption and nepotism dominating the KDP-PUK alliance in Kurdistan and has the support of the Peshmerga and academics) but their sister party (KDPP) survives in Syria and an ally in Iran (Komala Party), and KIU is another reformist party in Iraq that has significant representation and yet another is the KDSP pushing self-determination and not to be confused with the KDSP in Syria and is #2 there behind PKK (also in Turkey) & PYD...and a host of others. It's a mess. But, if ever granted their ancestral lands, they'd have to work that all out as well as their ultimate form of government...at this point I don't think there is a dominant political strain to point to.