As far as domestic American leftists at the turn of the century go, I think the most openly militant tendency at the time would've been found in the anarchists of the Pacific Northwest: the predecessors of Antifa, best known at the time for kicking off the 'Battle of Seattle' in 1999 against the WTO's conference in that city. So some especially radical, and certainly far better organized/competent/successful, ancom types are your likeliest domestic leftist candidate for the alt-9/11 IMO.
And as for the effects...well, there's not gonna be a War in Afghanistan beyond the Taliban-Northern Alliance conflict that was already ongoing, since with a far-left domestic terror cell executing 9/11 instead of an AQ sheltered by the Taliban there's no reason for the US to invade Afghanistan at all. There will probably be an Iraq War at some point though, since Bush and the neocons really wanted Saddam down & in the ground. Maybe they'll invade Syria, Libya or Sudan to compensate for missing out on Afghanistan - those were some of the other targets Wesley Clark alleged the Bush administration wanted to hit before getting bogged down by the Iraqi & Afghan insurgencies, and neocon warhawks have plenty of reason to hate them all.
Most of all though I think the War on Terror won't be 'brought home' so much as it will start at home. Seattle, Portland, Olympia and those other PNW cities known to be hubs of ancom activity are going to get turned upside down in the initial manhunt for the terrorists, then in the purges of anarchist activity to follow. I'd expect anarchist symbols & memorabilia to vanish as waving a red-and-black flag in public ITL will become as unacceptable as waving a black flag with the shahada & 'Death to America' on it IOTL, even in liberal cities, immediately after 9/11. The punk subculture mercifully dies outright instead of becoming a skinsuit for hipsters, who the original punks would frankly probably beat up for being unbearable, to wear in the 2010s. Fewer Muslims in Gitmo and a lot more 'gutter punk' types in high-security/military prisons across the country instead, probably.
Might see an embrace of 'proper' Communism, with no anarchist connotations whatsoever, among the far-left fringe instead - a lot more tankies and a lot fewer anarkiddies/Antifa wannabe streetfighter types, basically. If the economy still melts down around 2007-08, I'd imagine the Tea Party movement has a good chance of swallowing up most of the Occupy types (at least, the less ideological people who are just mad at the established system's failures and ruination of their finances) since the American people will still have a less-than-fond memory of what 'revolutionary terror' looks like in their lifetimes. In turn I'd also expect the Tea Party to grow well beyond the ability of the GOP's own establishment to control, shifting the domestic Overton window far to the right. The Democrats would have to very loudly disown their left wing to avoid being slammed as guilty by association, diving ever deeper into Clintonite neoliberalism.
While neoliberalism and neoconservatism lie discredited and slowly dying while anything to the former's left becomes unacceptable to the mainstream due to alt-9/11, American politics will likely come to be dominated by the fight between an empowered libertarian movement and properly reactionary nationalists who consider Eisenhower's tax rates, Taft Senior's tariffs and Taft Junior's isolationism to be part of the tradition they're restoring - both emerging from the Tea Party crowd. Basically early Trumpists in the latter's case, except presumably more coherent ideologically since they'll have some extra years to formulate, come together & go mainstream. I expect they will win out in the struggle for control over the Republicans since the national security/law & order mainstreamers will have an easier time allying with them than the libertarians, and they in turn would have fewer problems with coexisting with (and wielding!) the post-9/11 state security apparatus than the libertarians too. I could see said libertarians trying to take over the Democrats (whose socially progressive policies and recent embrace of economically liberal positions will make them much more amenable to a libertarian hijacking than the GOP's neo- and paleo-conservative tendencies IMO) instead.
Tl;dr this timeline's probably your best shot at getting a Mike Gravel (D) vs. Mike Huckabee (R) showdown in 2008, and will accelerate the switch of the working class in the Rust Belt to the Republicans that much more quickly.