I think much of their songs fall into the broader Outlaw Country genre, but yes bluegrass, folk, and yes a bit of rock. Also I have pet Pete Bernhard and I swear he is like a secret Maoist or something. Very cool dude. The ones I listed definitely have some very country bluegrass twang and bluegrass itself is sort of country subgenre.
But yes they are one of if not my favorite band of all time. A hell of a lot of class consciousness and marxist-adjacent critique of cops, landlords, even imperialism.
For example Poison Trees is a very clear critique of the entire institution of US policing and civil rights struggles, Never Learn is less clear but has obvious anti-landlord/enclosure of the commons/exploitation themes with a clear understanding that if we could unite we could bring the entire global economy to a halt to make our demands impossible to ignore.
Sweeping is just outright anti-imperialist.
And as for country, Beneath the Piano is possibly one of the most country songs of all time.
*Edit: LMAO met not pet, but that is too funny a typo so I'm leaving it.