Rothschild is one of the world's most august financial institutions, reflected in its discreet yet opulent new City HQ designed by Rem Koolhaas's OMA

The City of London is, in its own special way, surprisingly fond of architecture. You might have thought that niceties of design would get in the way of its relentless contest with other financial centres to be the most fearsome money machine in the world, but no. The rulers of the City permit themselves the incredible luxury, inconceivable in Singapore, Shenzhen or even Canary Wharf, of weighing and deliberating every tweak of its fabric.

There are the historic buildings, the monuments of Wren, Hawksmoor and Lutyens, that are reverentially coddled. There are also the monuments of the masters of our own time, as recognised by the biggest architecture award in the world, the Pritzker prize. There are works by no fewer than five winners of the prize (Norman Foster, Richard Rogers, Jean Nouvel, James Stirling and Rem Koolhaas's practice OMA) within the Square Mile. A sixth, Renzo Piano's Shard, makes its presence felt from just outside its boundaries. Such concentrations are hard to find outside places such as Saadiyat Island, the instant cultural district under construction in Abu Dhabi, or the 1980s tea services designed for the Italian company Alessi, by the biggest stars of the time.

The latest addition to the collection, OMA's whitish tower for the financial advisory group Rothschild, has ghosted its way on to the skyline with a surprising degree of discretion. Usually every sneeze of Rem Koolhaas and his team is the object of global fascination by architects and followers of architecture, but this not-small building has been sitting there for some time, its exterior more or less finished, without anyone paying much attention. Now the interior fit-out is also complete, bar a few details.

The discretion is part of Rothschild's corporate personality. As a distinguished 200-year-old institution, it doesn't feel the need to shout. It doesn't put its name on the door, and while it hangs a coat of arms outside, reused from former buildings, this is not very communicative to non-students of heraldry. It is located in a lane of extraordinary narrowness a short distance from the Bank of England, a narrow strip of pitted tarmac that seems one remove from being a cart track. You are supposed just to know that it is there and if you don't, you are not someone who needs to know or whom it needs to know.

You do, however, know that you are in the presence of something with a high degree of self-confidence. From the lane you rise through a steel colonnade to an ample podium of perfect emptiness, the main body of the building overhead, which then opens on to an also ample reception area. You are treated to the luxury of sheer space, precisely delineated with the oblong architecture. The floor is of travertine, also the ceiling, which creates a vertiginous blurring of up and down. Off to one side is an oak-shelved library that will house the Rothschild archive.

Should you be allowed past the security barriers you can then rise through the building, past the gym and cafe, and floors of close-packed desks, to the top levels of meeting rooms, dining rooms and events suites. There is a quasi-Soviet collectivism about the way the place is organised; as in the 1920s Narkomfin housing project in Moscow, the space allotted to individuals is modest, but the shared spaces of exercise, eating and meeting are generous.

In these spaces, an ever more magnificent panorama unfolds. In one direction St Paul's Cathedral sits in mighty repose, placed in the middle of a glass wall as if it were put there for the special benefit of Rothschild. In another there are the Gherkin and other towers of the City, which somehow look more impressive and serene than they do from ground level. These are celestial, Olympian spaces that convey the certainty that this – here, at this elevation, in this part of London – is where Rothschild belongs.

It is not all about sheer pomp and prestige. This is not OMA's way, and running through the building are touches of wit, irony and teasing. There is a play of small and big, which starts with the transition from lane to podium and continues with such things as extra-heavy or extra-light handrails. There are very thick walls ("Like castles and palaces," say OMA) and very thin ones made of glass.

There is also a play with the history of which Rothschild is so proud. In the meeting rooms are ancestral portraits, of well-mounted men riding to hounds and such like, and antique furniture. These are placed, with a touch of the eclecticism of a boutique hotel, alongside glass and aluminium, the latter embossed, in another moment of old/new overlay, with woodgrain patterns from the old oak panelling.

In Richard Rogers's Lloyd's Building a Robert Adam interior, imported from the institution's earlier premises, was recreated. There, it is a touch embarrassing in relation to the high-techery around it. In Rothschild the interplay of oak, oil paint, silk and aluminium is where all the fun is to be had. It delivers the required message that the institution is both ancient and modern. More than that, it is shown to be cultured, sophisticated, self-aware and sufficiently self-assured to allow a little humour. Rothschild advises but doesn't lend, which sets it apart from the casino banks of ill-repute, and its architecture reminds you of this fact.

OMA also likes to squeeze whatever public value there might be in a commission, even out of a discreet private bank. The colonnade along the lane can be used by anyone, in effect widening the street, and on the far side of the podium a view opens up to the churchyard of Wren's St Stephen Walbrook. It is clear that the podium is privately owned space, but the building still offers more than the many City blocks which rise sheer and opaque from the pavement. Next door, for example, one of Foster's least good works has been squelched on to the ground, an assertive, ribbed, over-inflated blob that is oblivious to its surroundings. OMA's building interacts with its neighbours, enriching itself and them in the process.

The City's fondness for architecture has, in fact, its limits. Often it runs as far as licensing a big name to sculpt the external form of a block, but not to such architectural qualities as the play of volumes and scale, the interconnection of outside and in or the creation of three-dimensional settings for the lives that go on in and around a building. Rothschild does all these things, with skill and subtlety. The only shame is that some of the best bits are on the far side of the security barriers. Come the revolution, though, it will make a great collectivist housing scheme.

An exhaustive survey of the work of Rem Koolhaas's OMA is as intelligent and challenging as the practice itself

Here east meets west, and old meets new, in an intoxicating fusion that assaults the senses. Everywhere you look exotic shapes rise out of the hustle and bustle of the modern city. It is a place of colour and contrasts, of bright lights and glittering glass, of up-to-the-minute technology and ancient wisdom, of chic and stylish fashion, cutting-edge design and exquisite craft, as well as delightful and original gift ideas for the discerning shopper who wants something a little bit different…

It is tempting to write a review of the OMA exhibition at the Barbican as a travelogue, because arrival there is very like entering a foreign city where you are somehow supposed to make sense of a huge and opaque mass of image and information, and of inscrutable objects. In this show, created by the Brussels-based design collective Rotor, there is more stuff – pictures, words, films, things – than anyone could ever absorb, including a 48-hour slideshow that flicks through all 3.5 million images currently stored on OMA's database. Only the exhibition shop, placed at a pivotal point in the show, with desirable objects placed cleanly on tables, offers the security and certainty that comes with buying something. Hah! A vase! I know what that is (even if its shape is based on a concert hall). I'll have that.

In its resistance to comprehension, the show resembles its subject, OMA, or the Office of Metropolitan Architecture, the Rotterdam-based architectural practice set up in the 1970s and now an organism too vast and intricate for anyone to know fully, including its leader Rem Koolhaas. I expect other offices – Foster's, say – produce equal or greater quantities of stuff, but they tend to head single-mindedly in the same direction. OMA's interests are many and promiscuous. They don't just design buildings; they also research, publish and speculate, and take on commissions such as redesigning the identity of the European Union, for which kind of work a subsidiary – AMO – was created. Yet a common intelligence runs through it all.

At the press view the thing that Koolhaas really wants me to look at is not the exhibition but a book he has just produced about Japanese architecture of the 1960s. OMA aspires to be a university as well as a design practice. They carry out their own architectural criticism: they once produced a publication revisiting some of their buildings after they had been in use for a while (something professional critics rarely do) and pointing out what did and did not work. Not to the extent, however, that the reader would cease to find them admirable.

An attraction of OMA is that they are less stiff-necked than other practices. They admit mistakes – the exhibition shows a marvellous film of a famous house they designed in Bordeaux seen from the point of view of its housekeeper, who has to lug a vacuum cleaner up a spiral of sharp-edged steps. They try to avoid piety and hypocrisy. They admit that they are subject to the whims of the clients and commercial interests that hire them, but "instead of whining about it, we developed an aesthetic about it". They have a line in self-deprecating humour, and the show features two enigmatic lumps from the firm's archive, reverentially lit as if artworks. "It is unclear to the archivist," says the caption, "whether these are models or just clay leftovers."

Koolhaas once said that "the market economy is considered the final and ultimate logic of not only economic life but also political life and increasingly also of cultural life", not approvingly but as a statement of fact. He and his practice also hold on to the idea that architecture might contribute something to public life, and to experiences that are not pure commodities. Their combination of realism and idealism means that their work is a constant operation of fancy footwork at a global scale, of outsmarting and outmanoeuvring the forces behind their commissions, at the same time that they understand them and harness their energy. They are constantly seeking a route between complicity and impotence.

In the Barbican exhibition their flirtation with the market means that they put the shop – usually treated as an embarrassing but compulsory addendum – in the middle of the show. They have fun with it, curate its contents amusingly and honour it with an impressive space. But they also open up a public route through the galleries through which, free of charge, you can pass, and see some of the exhibits. The show plays games with value, papering the walls with print-outs and the contents of office wastepaper baskets, while fully aware that the drawings and models of famous architects are now precious collectibles.

At a slightly larger scale their play of power and freedom is facing its greatest test with their Beijing headquarters for China Central Television, or CCTV, which is due to open next year. This is described as "the second largest building in the world", though by what measurement I don't quite know, and it houses and celebrates an organisation which, as the main communications arms of a dictatorship, is not exactly benign. OMA's argument is that it is better to engage with such things than ignore them, and their astounding-looking building contains such would-be positives as a public route through the interior. The as yet unanswered question is whether this will be enough to change CCTV in any significant way, or whether it will only add a sheen of sophistication to their operations.

Koolhaas, and OMA, are happiest when poised between different points of a paradox or an opposition. They work with authority, but their ethos is freedom. They are collaborative, and their name implies a kind of anonymity, but Koolhaas is the most distinctive and famous individual in his business, and everything in OMA emanates from him. The scale and global spread of their projects, and of their intellectual ambitions, is enormous, but they have a sort of modesty about what architects can actually achieve.

They continually yearn to be something more than architects – politicians, economists, ecologists, philosophers – but they always end up doing buildings, which is what they are best at. For all their voracious fascination with almost everything, especially things big, challenging and new, they can still produce habitable structures, like CCTV and like the simple but subtle Maggie's cancer centre in Gartnavel, Glasgow which opened last Monday. These buildings would be different, however, without the curiosity and research.

The exhibition shows it all: the searching, the proclamations, mishaps and follies, the achievements, the things of beauty and the detritus. It does not present, as some architecture exhibitions do, a series of projects to be contemplated and understood. Architecture is rather seen sidelong. A problem of architectural exhibitions is that their subject is the background of life, but must become foreground for the purposes of a show. In Rotor's installation it is still background, of a particularly vital kind.

It starts opaque and baffling, but hard work is rewarded by pleasure. It eventually unfolds into a display of fierce energy and intelligence. Just don't expect to understand or even see it all. Nobody will, not even the people who made it.