A man is judged by the company he keeps. A woman born in the nineteenth century, especially one who had ambitions as an artist, was more likely to be defined by company she was prevented from keeping. To rub shoulders with Mondrian, one of the revered practitioners of abstract art, is proof surely that Swedish artist, Hilma af Klint, who struggled to exhibit in her own lifetime, has finally been invited into the fold.

Hilma af Klint & Piet Mondrian: Forms of Life is a welcome departure from Tate Modern’s default format, the single artist blockbuster. Placing two artists together in visual dialogue raises the level of curatorial ambition and offers a more rewarding viewing experience, but it also carries risks. The aim evidently is to reveal parallels and intersections, and so redress an historical injustice, but the juxtaposition may end up exposing irreconcilable differences even where they seem most alike. Another danger is that it will be viewed crudely as a two horse race, which is predictably how most UK newspapers have covered the show. One review that I saw snarkily complained that Mondrian’s Broadway Boogie Woogie was not included, insinuating that one of the runners had been hobbled. In fact, we know that af Klint was handicapped in ways that Mondrian never was, purely by reason of her gender.

The first thing to be said is that this is a very intelligently put together show. It is thoughtful, and thought-provoking. At the same time, it is breathtakingly beautiful. Tate Modern’s brand is about embracing the spectacle and the instant gratification of what has lately been dubbed the experience economy. Think Yayoi Kusama. Here, I felt treated more like an initiate, or apprentice, on a path to enlightenment, rather than simply needing two hours of my leisure time filled. Some of the subtleties of the selection and hang only yielded themselves up on a second visit. An eye-catching early Mondrian landscape, for example, has a singular tangerine cloud hovering brilliantly in a pallid twilight sky lit up by a sun that one infers has already set. It took a while for the penny to drop and for me to see a connection, suggested but not spelt out, with the abstract ‘thought forms’ of Theosophists Annie Besant and Charles Leadbeater that similarly relay an invisible source of illumination.

Those illustrations, and much else, are on display in a central room dedicated to tracing a multiplicity of connections with a wider intellectual culture pertinent to each artist’s abstract idiom. It turns out they both had a background in scientific illustration. That was extremely consequential for Af Klint as can be seen from her exquisite botanical depictions of plants native to Scandinavia juxtaposed in a 1919 notebook with their abstract geometric projections onto an astral plane – a double-take in a minor register of her majestic series The 10 Largest. It is fascinating as well to see how readily she absorbed a microscopic language of cellular division at a time when the details of such a process were only patchily understood. In one of the works from her Eros series, the nuclear spindle that forms when cells replicate is stretched obliquely across the picture plane, a biological fact become pictorial metaphor. No less intriguing was to be shown how Mondrian’s illustrations of clumps of bacteria stained in order to be viewed under a microscope find echoes in early De Stijl abstract paintings.

Spiritualism can’t exactly be called an elephant in the room, since it is tackled head on, but it does raise urgent issues for how we respond to the work; how we value it, and define it — as art, or something else entirely. Those so inclined can compare the iridology charts and horoscopes of Mondrian with those of Af Klint. In Mondrian’s case, I felt one can either take the occultist baggage or leave it; with Af Klint it is more intrusive. Part of me wants to side with André Breton’s categoric rejection of crankish notions of dictation by spirits in an essay where he nonetheless acknowledges the derivation of surrealist automatism from the sorts of drawings Af Klint and her mediumistic colleagues produced en masse from 1896 to 1908. But, on the other hand, I can’t help thinking that ideas of inter-connectedness, of an immortal, indestructible soul, speak directly to our bathetic, social media-defined present. A fringy belief turned dystopic reality. What is cyber-space, after all, if not our immaterial ether?

One needs to remember also that religiosity and esotericism enabled the five women, later expanded to thirteen, who met for regular séances to opt out of an oppressive heterosexual matrix without having to make excuses. The copious minutes and personal diaries they left behind reveal that romance blossomed under the cover of a quasi-sacral environment replete with kitschy Rosicrucian altars and the like. Af Klint and Anna Cassel — the pair were probably lovers — adopted the nicknames askel (chaste) and vestel (virgin), code words that crop up repeatedly in pictures it is now thought may have been co-authored. Obedience to the Masters was their concession to a man’s world, though these spiritual guides turn out to be pliable, like an indulgent parent who ‘instruct’ you to do exactly what you want to. More difficult to comprehend is af Klint’s cultish regard for Rudolf Steiner. The lowered ambition of the wet-on-wet pictures made according to his dogmatic precepts is sad to see. Her grovelling letter displayed in a glass case to someone whose influence was nothing but destructive seems inexplicable, unless it was meant sarcastically.

In several rooms, the artists go head-to-head. I found myself constantly having to check dates because the visual-cum-thematic correspondences play havoc with an orderly sequence. Evolution, a pre-eminent paradigm in biology coopted by  Theosophists to describe an evolution from matter to spirit, is the explicit subject of major works by both artists. Mondrian employs the tripartite formula of a religious altarpiece. On a facing wall we witness the seriality that is Af Klint’s consistent modus operandi in the paintings executed for the Temple. This is succeeded by a room devoted to flowers and another room for trees. It was stunning to see Mondrian’s rarely-shown flower pictures together, those dated post-1921 scarcely distinguishable from the early much versions with stylised Art Nouveau signatures. Af Klint’s exquisite watercolours on an opposite wall transcend mere illustration, showing that she can hold her own in this exalted company. In the next room, one encounters an early masterpiece in Mondrian’s The Apple Tree (1912). The finely calibrated aesthetic decisions evident in the revisions and over-painting are a rarer pleasure in Af Klint, but her work offers other rewards. A fourth room in this exhibition full of surprises draws the two artists together around their use of the colour pink. A note appended to one of the pictures in The 10 Largest series shows that af Klint employed an organic pigment, rose madder, extracted from the common madder plant Rubia tinctorum. This colour derived from the plant world serves her in the series Eros (whose creation immediately precedes those just mentioned) as the vehicle for a radical female Eros unprecedented in high art.

At risk of oversimplifying, in Mondrian one observes a stepwise march toward the non-figurative that with hindsight has an air of heroic inevitability. It is an evolution internal to picture-making that clearly begins before his encounter with cubism: a reminder of the crucial but often overlooked importance of late nineteenth century symbolist aesthetics for the modernism that followed. By contrast, in Af Klint’s oeuvre, perplexingly, abstraction seems to arrive fully formed and to coexist with elements of figuration, writing, and abstruse symbols. Her commitment to the abstract seems altogether more capricious. Mondrian’s is marked by a radical reductionism; she embraces complexity and contradiction. Mondrian’s linear progress to geometric abstraction leaves his flower pictures in an odd place, as an unassimilable remainder. This isn’t so with Af Klint: botany suffuses her abstraction. Mondrian unproblematically endorsed a male-vertical versus female-horizontal equivalence; the leitmotif of her work is a ceaseless struggle to overcome the tyranny of gender binarism. This last point makes me leery of tagging her work with the loaded descriptors ‘nature’ and ‘the natural.’ Was not her task rather to de-naturalise gender essentialism through constant reference to a non-human world of plants?

Af Klint’s series The 10 Largest (1907) rightly culminate the exhibition. I’ve lost count of how many times, and in how many different countries, I’ve seen these pictures in the last decade. One hopes that having done their job the welfare and longevity of what are fragile works on paper will now come first. At Tate Modern, they are shown in a darkened, contemplative, chapel-like room, a makeshift Church of the Holy Sepulchre. Af Klint’s ‘vegetal universe’ — curator and scholar Briony Fer’s apt phrase – achieves a grand apotheosis in this series. They alone ought to guarantee their creator a prominent place in the art history of the early twentieth century. It is astonishing that they were painted in the same year as Picasso’s Les Demoiselles d’Avignon and a good ten years before Mondrian’s mature De Stijl abstraction. For exuberant organicism and coloristic pizzazz they rival Joan Miró at his best. The expansive scope of the imagery, extending from the microscopic to the cosmic in scale, is like a textbook illustration of Burke’s sublime. I don’t profess to fully understand them – that, after all, is the nature of esotericism — though Julia Voss’s thoroughly researched biography and assorted volumes edited by Kurt Almqvist and Daniel Birnbaum that have appeared since 2013 have begun to unveil the Mysteries.

When I proposed a connection with the biological notion of vitalism, a concept that bridges science and spiritualism, I was largely unaware of the intricacies of Af Klint’s all-female milieu. Were I to approach the subject again, I would want to emphasise a linkage between plants and queerness. Briefly, the taxonomic system introduced by the Swedish botanist Carl Linnaeus based on counting the pistils and stamens of flowering plants revealed to the eighteenth century a licentious plant world saturated by eroticism, as Erasmus Darwin’s poem “The Loves of Plants” (1791) abundantly attests. Linnaeus’s ‘sexual system’ revealed to a scandalised public that hermaphroditism is the norm in the plant world. A century later, in fin-de-siècle France and elsewhere, plants once again entwine with sex in the emergence of a homosexual identity; as seen, for instance, in the fondness of Oscar Wilde and his circle for white lilies, and in such era-defining literary works as J.-K. Huysmans’ Against Nature (1884) and Octave Mirbeau’s The Torture Garden (1899). Af Klint’s artistic project, I believe, takes root in this nutritive humus, as much as in the ubiquitous organicism of Art Nouveau.

The Freudian notion of ‘deferred action’ (Nachträglicheit) is helpful in thinking about the untimely timeliness of Af Klint’s oeuvre. After untold disappointments, she concluded the world wasn’t ready to receive her message. The Hilma af Klint Foundation has dedicated itself tirelessly to ensuring she attains her rightful place inside the art historical canon. They will see this show as mission accomplished. Others, equally well-intentioned, would prefer that she remain an outsider to the art world, unsullied and inviolable. No one would now dare suggest that her work is somehow not art, but maybe ‘abstract’ is an ill-fitting straitjacket, insufficiently capacious. I see a cautionary tale in the modernist critic Clement Greenberg’s obviously tendentious claim that Miró was an abstract artist. Some have seen ‘diagrammatic’ as a better description of it. I find myself rather persuaded by that. Her pictures are like logical propositions in paint, not dissimilar to post-Bauhaus Kandinsky. All in all, we urgently need a new toolkit for talking about images that recognises the artwork, abstract or not, as information, an array of data. So, is she or is she not a pioneer of abstraction? This superb exhibition allows you to make up your own mind.

Hilma af Klint and Piet Mondrian: Forms of Life is at Tate Modern, London to 3 September.