A pun is a word that doesn’t mean what it says, or rather, means what it says but also means something else. It is a signpost bearing the same destination, but pointing in two directions.

The longest pun-free period in Duck Soup is at the beginning, after the unsettling opening shot of ducks paddling in a cauldron over a hot fire. We are in the majestic council chamber of the government of Freedonia. A meeting is in session. Zander, the president, is asking the wealthy Mrs Gloria Teasdale, widow of the late Chester V Teasdale, for a further $20m, so that he can announce an immediate reduction in taxes. Mrs Teasdale, played by the redoubtable Margaret Dumont, complains that she has already donated half her fortune, and will only lend more money if a new leader is put in place.

It is already a minute in, and there is still no pun in sight. Instead, the language of negotiation fills the air: this could be any political film, at any time, dealing with the usual problems of borrowing, taxing and spending. You almost feel that, if you looked very hard, you could spot George Osborne or Vince Cable in the background, beavering away.

“In a crisis like this, I feel Freedonia needs a new leader,” declares Mrs Teasdale. “A progressive, fearless fighter, a man like … Rufus T Firefly!”

Firefly is, of course, Groucho Marx, but we have still not had a glimpse of him. Instead, the scene switches to an absurdly lavish ballroom, its vast staircase flanked by overly plumed guards and trumpeters with exceedingly long trumpets. It is the reception for His Excellency Rufus T Firefly, who has yet to arrive. After various formal introductions, a troupe of ballerinas enters, scattering flowers along his projected walkway. Everyone choruses:

His Excellency is due

To take his station,

Beginning his new administration

He’ll make his appearance when

The clock on the wall strikes 10!

We’ll give him a rousing cheer

To show him we’re glad he’s here!

Hail, hail Freedonia!

Land of the brave and free!

In 1933, when Duck Soup was first released, this deadbeat song with its patriotic lyrics would have brought to mind the forced jubilations of Stalin’s Soviet Union or Hitler’s Germany; even today, it remains the prototypical song for countries such as Azerbaijan and North Korea, speaking of order and shared purpose, of a population pointing in the same direction. And still no sign of a pun.

The minutes tick by, and the ceremony keeps on stopping and starting as His Excellency fails to appear. All eyes are on the entrance when Groucho Marx finally lollops in from the side, unseen. Finally, he is greeted – “I welcome you with open arms” – by a clearly besotted Mrs Teasdale. The moment he enters, the puns come thick and punishingly fast, with no letup. The more Mrs Teasdale coos, the more he undermines her with his crazed wordplay. “You can leave in a taxi. If you can’t leave in a taxi you can leave in a huff. If that’s too soon, you can leave in a minute and a huff.” These quick-fire speeches have a topsy-turvy logic all their own, each word changing its meaning the moment it is uttered.

Everything about Groucho was a pun. His father was called Simon, not Sam, and Marrix, not Marx; his mother’s real name was Miene Schönberg, but her stage name was Minnie Palmer. Groucho was born Julius; he only transformed into Groucho in a break between shows, when a fellow comedian allotted nicknames to each of the Marx Brothers: Milton wore rubber boots, hence Gummo; Leonard chased women, or “chicks”, hence Chico, and Adolph played the harp, hence Harpo. And as for Julius, he carried a small drawstring bag known as a “grouch”.

Groucho’s most visible characteristic is itself a sort of pun. It is both a moustache, and not a moustache. Instead, it is a painting of a moustache – ceci n’est pas une moustache – that he first daubed on his face with greasepaint in 1921, having arrived late for a performance, with no time to glue on his artificial whiskers. Before long, his eyebrows followed suit. Between them, they were to become the great symbol of the Marx Brothers, loved by low-brows and high-brows alike.

The world of Freedonia is a world built on order. Everything is where it should be and does what it is meant to, before the Marx Brothers send it all haywire. After their arrival, nothing is allowed to remain in its rightful place, or to be what it is meant to be. Everything is transformed into something else. The world has become a pun.

Amid much bugling, successive guards announce “His Excellency’s car!” But even the car is not a car: following another fanfare, Harpo appears on a motorcycle. “I’m in a hurry,” cries Groucho, stepping into the sidecar. “To the House of Representatives! Ride like fury. If you run out of gas, get ethyl. If Ethyl runs out, get Mabel. Now, step on it!” And with that, Harpo accelerates off, leaving the sidecar firmly in place. “Well, it certainly feels good to be back again,” says Groucho, having got nowhere.

When Harpo next appears, he is with Chico. They arrive at the Ambassador’s door as his appointed spies, both wearing bearded masks and hats, with Harpo’s eyes whirling round on springs. But the mask itself is then unmasked: Chico turns Harpo round; his real face is on the other side of his head. The pair are double agents, or human puns.

And so it goes on, order overturned, conformity unravelling, idiocy at the gates. Later – I was about to say much later, but the whole film lasts barely an hour and 10 minutes – we are in a courtroom, with the two spies on trialfor selling Freedonia’s secret war code and plans. “Sure. I sold a code and two pair of plans,” chips in Chico. Under the pressure of the courtroom, the puns bubble over, particularly when Chico turns turtle on the prosecutor:

Chico: Now, I ask you one. What is it has a trunk, but no key, weight 2,000lb and lives in a circus?

Prosecutor: That’s irrelevant.

Chico: A relevant! Hey, that’s the answer. There’s a whole lotta relephants in a circus.

Judge: That sort of testimony we can eliminate.

Chico: ’At’s-a fine. I’ll take some.

Judge: You’ll take what?

Chico: Eliminate. A nice cold glass eliminate.

Firefly: Gentlemen, Chicolini here may talk like an idiot, and look like an idiot, but don’t let that fool you. He really is an idiot.

One pun leads to another: they bounce with the speed and precision of a polished stone skimmed across a lake.

Minister of Finance: Something must be done! War would mean a prohibitive increase in our taxes!

Chico: Hey, I got an uncle lives in Taxes.

Minister of Finance: No. I’m talking about taxes – money, dollars.

Chico: Dollas! That’s where my uncle lives. Dollas, Taxes.

It is the rat-a-tat-tat speed of their delivery that distinguishes the Marx Brothers from their contemporaries, and stops them growing old. Other comedies from that era now seem sluggish and laborious: Duck Soup is alive and kicking.

Harpo, was, of course, the silent one, but the scenes in which he appears are generally far louder, and more perilous, than any of the others. He is a bundle of anarchic energy, targeting the pompous and the self-regarding, like all the vengeful characters in Struwwelpeter rolled into one. Always taking out his scissors and going snip-snip-snip, he snips off the ends of Chico’s frankfurter and Firefly’s quill pen, he snips off the peanut vendor’s apron, the ambassador’s tail coat and the soldiers’ fancy plumes. Instead of speaking, Harpo grins like a madman and honks his horn. Having broken into a house, he tries to open the safe, but he twists the volume dial of a radio by mistake. It starts to blare, so he has to smash it. Whenever he is on screen, even the most inanimate of objects cannot be trusted to stay quiet. At one point, he shows Groucho a tattoo on his stomach of a house. As Groucho takes a closer look at the tattoo, a dog pops out of it; when Groucho says “Miaow!” the dog starts barking.

The most celebrated scene, and the most sublime, revolves around another visual pun: a mirror that is not a mirror, a reflection that is not a reflection. Harpo, disguised as Groucho, tries to fool Groucho into believing that he is looking at himself in a mirror. Groucho, suspicious, executes a random series of increasingly ludicrous movements in the hope of catching him out. But whatever movement he makes, however daft, Harpo anticipates him and does just the same. Even when Groucho walks into what should be the mirror, and out the other side, Harpo does the same, but in reverse. Eighty years on, that visual gag is still going strong: in an episode of the gloriously perverse American cartoon series Family Guy, the baby-man antihero Stewie is transported back to 1930s Berlin, where he disguises himself as Adolf Hitler. Then Hitler himself comes into the room, and Stewie has to convince him that he is looking into a mirror.

The world in 1933 was becoming increasingly overpopulated with tin-pot dictators. Is Duck Soup a political satire? Many intellectuals and academics have argued as much, though they tend to belong to the school of criticism that likes to stamp all the jokes out of comedy so that they can praise its high seriousness without fear of contradiction. A New Yorker cartoon once showed a Professor of Semiotics saying “The tautology of their symbolism thus begins to achieve mythic proportions in A Day at the Races, Duck Soup and A Night at the Opera”. As is so often the case with New Yorker cartoons, it veers from reality only by a couple of degrees.

Over the years, Duck Soup has been praised for its understanding of paranoia in international diplomacy and of the economics of warfare. It is full of gags about the futility of war and its financial advantages. At one point, Groucho is told of a peace plan and replies, “It’s too late, I’ve already paid a month’s rent on the battlefield.” At another, Groucho puts his hand on Harpo’s shoulder and tells him, in ringing tones, “You’re a brave man. Go and break through the lines. And remember, while you’re out there risking life and limb, through shot and shell, we’ll be in here thinking what a sucker you are.”

This joke was echoed, 30 years later, in Beyond the Fringe (“I want you to lay down your life, Perkins. We need a futile gesture at this stage”) but, taken as a whole, Duck Soup is too explosively nonsensical to let itself succumb to the discipline of satire, which might loosely be described as comedy with an agenda. (Nonsense is not an alternative version of sense, but rather its negation.)

On 14 November 1933, three days before the release of Duck Soup, Harpo took a ship to the USSR, via Hamburg. He was to be the first American artist to perform in Moscow since the US, under President Roosevelt, had recognised the Soviet Union. Without realising it, he was about to enter a real-life looking-glass world, where draconian decisions are taken for all the wrong reasons.

In his otherwise delightfully breezy autobiography Harpo Speaks, he says that in Hamburg he saw “the most frightening, most depressing sight I had ever seen – a row of stores with Stars of David and the word ‘Jude’ painted on them, and inside, behind half-empty counters, people in a daze, cringing like they didn’t know what hit them and didn’t know where the next blow would come from”.

On the train out of Warsaw, he bumps into a fellow American, who tells him he’ll have to pay for excess baggage at the Russian border, and kindly lends him a hundred roubles, saying it will be cheaper to pay in roubles than dollars. But when Harpo offers roubles to the Inspector, mayhem ensues. “Bells rang. Buzzers sounded. Boots clomped all over the place as guards came running.”

Harpo is hauled off to a shed, where he is cross-questioned. “Where did I get the rubles? A guy on the train lent them to me. What was his name? I didn’t know his name. I was lousy at remembering names. I was lying, the Russian colonel said. Tell the truth now: where did I get the rubles? I gave him the same answer.

“A squad of guards lugged my trunk and harp into the shed. ‘Open the trunk, please,’ said the officer. I unlocked it and the Russians began unloading it. When at first they only found a raincoat and an assortment of pants, shirts and ties, they were obviously disappointed.

“Then they hit the jackpot. From the trunk they removed 400 knives, two revolvers, three stilettos, half-a-dozen bottles marked POISON, and a collection of red wigs and false beards, moustaches and hands. More bells rang. More buzzers sounded. Whistles blew. More officials and more guards came clomping into the shed.”

Harpo is grilled. “Would I please explain why I was transporting weapons and disguises? I told them they were all props for my act. Act? What act? I said I had come to Russia to put on a show. Americans do not entertain in Russia, they said. I had better tell the truth.”

They ask what is in his harp case. A harp, he tells them. They order him to play something. “This would have been my salvation any other time, any other place, but in an open shed when it was 30 below zero. I was so stiff from the cold that I couldn’t get my gloves off. All I could do was run my gloved hands up the strings a couple of times and pray that somebody there would recognise the professional touch.”

But no: a guard gets the same noise out of the harp. The officials shake their heads and smirk. Harpo fears imprisonment, or worse, and begins to yell about his rights, but all to no avail. It is only when his fellow American pops up on the scene, and they accept his explanation, that Harpo is allowed to board the Moscow express. “The train was unbelievably crowded, ten and twelve people to each six-passenger compartment, and it stank of disinfectant, but I thanked God I was lucky enough to be on it.”

His show is a terrific hit with the Russians, who love clowns, but, on the way back to America, Harpo can’t shake off the awful feeling that for the past eight weeks he has been watched wherever he goes, “by eyes I couldn’t see. I never, not for a minute, felt I was really alone. I was a stranger who had stumbled into a deadly conspiracy, who had to be kept from finding out what the plot was all about.”

But one thing above all strikes Harpo about the USSR. “I never saw anybody do anything just for the hell of it. I never saw anybody pull a spontaneous gag.” Rationality, he feels, is the only acceptable justification for everything in Stalin’s USSR: something can only be funny if it has a reason to be funny. And this is why Duck Soup is so funny: because it has no reason to be funny.

• The Marx Brothers film season begins at BFI Southbank, London SE1, on 14 January. bfi.org.uk.