I emerge from the shady interior of my apartment block in the Università district of Naples and head up Via Mezzocannone, donning dark glasses to dilute the blinding sun – but I can’t blot out the spectacular display of rubbish decorating the street.

Garbage – along with sex, death and superstition – is an enduring Neapolitan theme. In the infamous refuse collection strike of 2008 apocalyptic piles of rubbish blocked these streets. Recycling bins are dotted about now, but some folk in this bohemian area still dispose of their crap in the traditional way.

Today’s display features a toilet basin with a water container and beer bottle stuffed inside it; half a dozen boxes of empty cans; a stroller minus one wheel; a large wooden pallet; plastic bags spilling unidentifiable filth; a dining chair with legs that look like they’ve been chewed; some sort of punctured inflatable; and for good measure an actual dustbin overflowing with nastiness.

Who’s to blame for all this? Odysseus, it seems. Because when he thwarted the attempts of the Sirens to seduce him with their songs, they threw themselves into the sea near here, and one of them, Parthenope, washed up on the Neapolitan beach and died – just after laying a sacred egg.

I know – weird stuff, but the Sirens were originally half-woman, half-bird (only later were they depicted as mermaids) and that egg was allegedly the beginning of a city that does weirdness to the extreme. Parthenope’s tomb was built where she laid that egg, and Neapolis was sited around that tomb by Greek settlers (it was a long time before it became Napoli and Italian).

Odysseus was also to blame for bringing me here a couple of years ago. As part of a trip following his wanderings, I explored what the Italians call their Odyssean Coast. Travelling from the Strait of Messina (location of Scylla and Charybdis) to the water-filled caldera of Lake Avernus (entrance to the Underworld) and Monte Circeo (lair of sorceress Circe) I had to pass through Naples.

I fell in love with it at first step. The narrow streets from Napoli Centrale to Montesanto (I had to walk from one station to the other) were a circus, a fairground, a musical, a riot, a food festival, and an obstacle course where pedestrians, scooters and three-wheel trucks all push their way through the mayhem and no one – and everyone – has the right of way. It’s irresistible chaos.

Now where Parthenope died stands rock-solid Castel dell’Ovo (Egg Castle), originally built over a tunnel in which Virgil, in the first century BC, had left an egg (in memory of that Siren egg). And when today’s “new” castle was built in the fourteenth century – you guessed it – another egg was placed beneath it.

All these eggs – symbols of fertility and rebirth – have long since disappeared into the omelette of history. But Parthenope and the Sirens are still here.

Not far from that garbage exhibition I pass my favourite Siren tribute statue at what is nicknamed Fontana delle Zizze (Tits Fountain) – it features a winged, feathered Parthenope squeezing her breasts to produce jets of water. The quenching liquid rains down on a model of Vesuvius – did I forget to mention that this city sits right next to a gigantic volcano and could at any moment go the way of Pompeii? The daily possibility of untimely death is part of life here. Parthenope, though, is here to protect us and in the meantime, I think as I walk past a grotesque hustler effigy beckoning me into a lottery office, fortunes can be won.

In our dreams? Well yes: you can buy a dream codebook here called La Smorfia which gives you your lucky numbers. Blood is 18, death 47; then there are certain sexual acts (68, 69), pudenda (6), buttocks (16), breasts (28) and assorted male bits (29, 30, 34).

I walk on past real-life hustlers to Napoli Centrale to meet my friend Zaza. I’m inured to most beggars, but I do give a few euros to an enterprising panhandler who sweeps up litter on Corso Umberto I in return for donations.

Zaza lives south of Rome but is coming to spend a few days with me in Naples, which she has avoided, put off by stories about its dark side. Local journalist (and man hiding from the mob) Roberto Saviano’s Gomorrah exposé of organised crime must have repelled many. But I aim to show her that it isn’t all drive-by shootings, bloody torture and drug wars.

As I enter the station a shamelessly naked woman is washing her hair surrounded by frolicking dolphins. It’s another, more modern Parthenope sculpture. A full-frontal Parthenope also presides over palm-filled Piazza Sannazzaro west of here, but the most erotic Siren fountain I know is at Maratea, down the coast – water running over erect nipples and a taut body, naked except for a barely-there bit of rope, arms raised in ecstasy.

Today more restrained depictions grace the Sirens exhibition at Museo Archeologico Nazionale. I stand entranced before local sculptor Lello Esposito’s contemporary take on the legend: a lifesize Parthenope, nude (of course), fish-tailed and for some reason armless. Zaza meanwhile plays with the new hologram of a Siren who sings requests – the scantily clad, red-wigged, blue-painted performer in the glass box is actually pop singer Francesca Fariello offering a menu of classic Neapolitan ballads.

For a break from Sirens we visit San Gaudioso catacombs where the remains of medieval skulls are still fixed in the walls, and hit the Cathedral to see the bones of San Gennaro, the city’s patron saint. Some of his blood is kept there too but only brought out on special occasions.

Later, on the way to the seafront, we pass a coach from local tour guide company Syrenbus (motto “Pleasure on Wheels”). The moon rises as we turn off Via Partenope, cross the bridge to Castel dell’Ovo, and settle in a bar overlooking the Gulf of Naples, gateway to trade and trafficking of any item you can think of, legal or not.

Naples is short of nothing, especially the bizarre. Like the shrine to Maradona whose centrepiece is a hair from the football god’s head. Or the Museo di Anatomia Umana which presents all sorts of real bodies and their bits – including the sexual organs of a hermaphrodite, and a horrifying range of deformed foetuses.

Zaza prefers to see some modern art and I tell her I’ve heard the Hermann Nitsch Museum is interesting but I haven’t checked it out.

Next day we visit this backstreet gallery and are the only punters. I’m standing over what appear to be blood-stained pieces of wood and canvas when Zaza nudges me into looking at a video above us. We watch a succession of naked young men and women led ritually before a crowd to take up crucifixion poses and have jugs of blood force-fed into their mouths or poured over their genitals.

This is part of Nitsch’s decades-long installation, his Orgies Mysteries Theatre – a mixture of sacrificial bodies, pseudo-religiosity and lashings of animal blood. The red stains at my feet are one event’s leftovers, themselves now artworks. I feel my attempt to persuade Zaza that Naples has nothing to be alarmed by is dissolving…

The Austrian Nitsch, who recently left for the big installation in the sky, aged 83, more than once had his work accused of gross indecency but found a home for it here in the city of blood, sex and death. It embraced him – just as I think he would have clasped all the Sirens fearlessly to his breast.

See Naples and die? No, seize Naples and live!

Nigel Summerley is a writer and editor who has worked for The Sunday Times, The Times, The Daily Telegraph, The Independent, The i, The Guardian, The Express and the Evening Standard. He has a serious Homer’s Odyssey habit and recently collaborated on a new guidebook to Ithaca called Walking in the Footsteps of Odysseus.