If guitar music, as we’re told, is entering its niche-interest jazz phase, rap is having its psychedelia moment. Experimentation is the norm, boundaries are tumbling, anything goes. And if Frank Ocean, Kendrick Lamar, A$AP Rocky and Future are the era’s fab four, and Kanye West increasingly its Syd Barrett, Run the Jewels must surely be the Monkees. A pre-scripted collaboration between Killer Mike and El-P with a cartoonish comedy aesthetic – this is the duo who tempered their darker second album Run the Jewels 2 with a remix album that used only cat meows for beats – they are nonetheless creating some of the most direct, masterfully crafted and memorable music of the age.

As the pair arrive, miming to Queen’s We Are the Champions beneath two huge inflatable zombie hands making the band’s trademark gun-and-fist sign, their double-act dichotomy is instantly apparent. Mike is the old-school master, hyper-rapping boastful verses laced with jabs at societal ills; El-P is his comic foil, with hints of the Beastie Boys’ puerile juvenilia. While Mike details his cocaine-dealing past during Legend Has It, El-P suggests he has a unicorn’s horn for a penis and would like to ejaculate on to our soft furnishings. Between tracks, he launches a solo poetry career with deathless lines about his formidable cunnilingus skills, and tells us how he’s smoked Camden clean of weed. “My dick got a Michelin star,” he claims during opener Talk to Me, which must rather take the shine off it for The Ledbury.

Their cool-cop/goofball-cop act is perfectly, and powerfully, weighted. When RTJ declare themselves “a missile to turn this whole town into dust”, you almost believe them. Even a nuclear event would struggle to make a deeper impact than Talk to Me or Legend Has It, built on bass whoomps that must be waking up creatures on the floor of the Mariana Trench. Whether firing out hopscotch raps with attack dog intensity over the Arabian rhythms of Call Ticketron or revelling in the southern voodoo groove of their DJ Shadow collaboration Nobody Speak, there’s a hammer-blow force and clarity to their delivery that packs a punch most rap gigs simply don’t have.

Like their three albums to date, the show grows increasingly political. Hints of anti-consumerism (Stay Gold) and US imperialism (Don’t Get Captured) creep in and by Lie, Cheat, Steal and A Report to the Shareholders they’re gunning all-out for “the stinking asshole of the apocalypse that our country has become”. El-P decries “y’all demons” who “talk clean and bomb hospitals” and Mike predicts the third world war unless we “kill our masters and start again”. After a balcony-quaking Close Your Eyes (And Count to Fuck), Mike makes a moving, impassioned speech about how defying our governments with simple human unity will help us “overcome this insanity”. Joking aside, RTJ are the hip-hop clowns becoming politico princes.

• At Field Day, London, 3 June, then at Summer Sessions, Glasgow, 24 August.