Luxembourg is a capital you read downwards.

01 A city of three heights

The confusion starts within the first ten minutes. You come out of the station, walk north through a perfectly ordinary nineteenth-century district of banks and hotels, and then the pavement simply stops having a far side. What is in front of you is not a square or a park; it is air, and then, a long way down, roofs. A city you had assumed was small turns out to have a second one underneath it.

02 The rock did the rest

The geography is unusually simple and it explains everything else. Two rivers meet inside the city. Between and above them stands a sandstone promontory with near-vertical faces on three sides, so that anyone holding the top of it only had to fortify the one side where the land continued. That is the whole reason a settlement of no great agricultural or commercial advantage became strategically valuable enough to be fought over by one European power after another for three centuries.

What is easy to miss is that the valleys were never empty. Settlement down there is older than the fortress above it: Frankish colonists were living along the Alzette long before the rock was built on, and by the tenth century there were already dozens of inhabitants in the valley. The lower town is not an overflow of the upper one. It is the original, and it was left below when the important business moved up.

03 Seventeen kilometres on the inside

Because the promontory is rock, the fortifications did not only go up. They went in. Successive engineers hollowed out the cliff into a network of galleries, gun chambers and stairways that at its greatest extent ran to some seventeen kilometres, with around forty thousand square metres of usable shelter carved into the stone.

The detail that turns this from a curiosity into an argument is what happened to those tunnels in the twentieth century. They were not a museum piece by then; they were infrastructure. During the two world wars the galleries served as shelters, holding as many as thirty-five thousand people. A defensive system designed against cannon was still doing defensive work four hundred years later, for the same city, under a completely different kind of attack.

04 The rampart became a promenade

From 1867 the fortress was dismantled — the walls, the outworks, the gates. This is normally the point at which a fortified city loses its shape and becomes an ordinary one, with a ring road where the defences used to be. Here it produced the opposite result, because the thing that could not be demolished was the drop itself.

So the fortifications were not replaced; they were repurposed. The surviving stretch of rampart along the northern cliff became a pedestrian promenade — a street whose one side is a row of houses and whose other side is a parapet with seventy metres of nothing behind it. The cleared firing ground outside the walls became an open square. The gorges themselves became parks. In 1994 the old quarters and what remains of the fortifications were inscribed by UNESCO; the protected perimeter covers 138 hectares, about 2.7 per cent of the city's area.

This is the specific pleasure of the place, and it is worth naming precisely. In most historic cities the defences are an object you go and look at. Here they are the surface you are standing on. You walk the fortress rather than visiting it, and the view it was built to command is now simply the view from the street.

05 The bridges are the only straight lines

A city cut by two seventy-metre trenches has an obvious problem, and its answer is a set of very large bridges. The Passerelle, the older of the two great viaducts, opened in 1861 and steps across the Pétrusse on tall arches. The Adolphe Bridge, finished in 1903, spans the same valley in one enormous stone arch. Below both of them there is no water to speak of and no traffic — just a wooded park at the bottom of the gorge, with paths.

That inversion is worth sitting with. In a normal city the bridge is the boring bit and the thing underneath it is the event. Here the bridge is at the level of daily life — traffic, trams, people going to work — and the event is underneath, in a green trench that most of the people crossing above never go down into.

06 How to walk it

The mistake is to treat the upper town as the city and the valleys as an excursion. Done that way, Luxembourg is a two-hour capital and people leave saying there is nothing here. Done vertically, it is a full day and then some, because every descent changes what you are looking at: from the promenade the valley is a model village, and from the valley the promenade is a cliff with houses on it.

So go down at least twice, and use different means. There are staircases cut into the rock, a road that switchbacks, a free public lift that drops sixty metres in about thirty seconds, and a funicular. Nothing is far: the whole thing is a city of around 137,700 residents, and you cross the historic part on foot in half an hour, most of it spent on stairs.

One last thing to notice from the top, because it completes the picture. Across the valley on the far plateau stand the glass towers of the newest quarter, built from the 1960s onwards for institutions and banks. It is a fourth level, added a thousand years after the first, and it obeys the same rule as everything else here: it is separated from the old city not by distance — it is barely two kilometres away — but by a seventy-metre hole in the ground.