Despite the exotic name and her equally exotic appearance, French-born, New York-raised artist Niki de Saint Phalle could have been a Glaswegian.

Like a gloriously gallus bird of paradise, she strode across the art world of New York and Europe and back to America's west coast from the early 1960s until her death in 2002 at the age of 71.

Even if you don't recognise the name, you may well have clocked her beautiful adornment of Glasgow's Gallery of Modern Art (GoMA), the neoclassical city centre building which is hosting this new exhibition of her work. Saint Phalle was commissioned in 1996 by Julian Saplding, former head of Glasgow Museums, to create the mirrored entrance to the gallery and the triangular mirror mosaic which is installed above the exterior columns.

On closer inspection, the mosaic depicts elements of the story of St Mungo. But as Saint Phalle, famed for feminist statements in her art, is the woman behind it, the mosaic incorporates elements of adultery and betrayal, which are a lesser known side of the Dear Green Place's birth story.

This new exhibition, which will run for 15 months, is quite simply a gift to the people of Glasgow. Glasgow Museums owns two important works by Saint Phalle, which the artist herself gifted to the city following a retrospective at the McLellan Galleries in 1993. The works, The Great Devil and Altar To A Dead Cat, are both on show here alongside 15 new works that have been donated by Eric and Jean Cass through the Contemporary Art Society. The £1million gift is the largest of its kind since GoMA opened and makes the city's collection of Saint Phalle's works one of the largest in the country.

Eric and Jean Cass, who are in their 90s, have dedicated more than 35 years of their lives to supporting artists. During this time they have built up a collection of more than 300 colourful modern and contemporary works by major names in 20th-century art, all of which were previously housed in the interiors and gardens of Bleep, their modernist 1950s home in Surrey.

Recently they decided to disperse the collection to museums across the UK through the Contemporary Art Society, and because Eric Cass knew Glasgow Museums owned works by Saint Phalle, he suggested the Society make an approach. The gift consists of 13 sculptures, one lithograph and related ephemera by Saint Phalle. They have also gifted two oil paintings by John Bellany, whose retrospective opens this weekend in Edinburgh.

Saint Phalle's work in this exhibition ranges from angst-ridden to joyful in one fell swoop, although this self-taught artist had distinctive periods in which she created one and then the other. One of the most striking is Altar To A Dead Cat, (1962), which is on show in GoMA in a little sequestered space all of its own. This powerful work belongs to Saint Phalle's Shooting Pictures period in which she worked out the internal demons associated with the sexual abuse she suffered at the hands of her strict Catholic father at the age of 11.

She explained it this way: "In 1961 I shot at daddy, all men, small men, large men, important men, fat men, men, my brother, society, the Church, the convent, the school, my family, my mother, all men, daddy, myself, men. I shot because it was fun and gave me a great feeling."

From 1960 to 1963, while still living in New York, Saint Phalle made her tirs, or shooting paintings. She embedded pockets of coloured paint into thick layers of white plaster on canvas, which exploded when she shot at them. The mixed-media Altar has been restored for this exhibition, and is harrowing and curiously exciting all at once, not to mention oddly beautiful. Be aware: devout Catholics and cat-lovers might be offended.

In the main body of this exhibition, we see Saint Phalle's work in all its characteristic brightly-hued, chunky glory. Her big "nanas" – voluptuous, bright, confident and often pregnant – jostle with other trademarks, such as serpents, snakes and dragons. Nothing is as it seems in a Saint Phalle artwork. Even her dragons have two faces: one jolly, the other snarling.

I left this exhibition half in love with this exotic creature who modelled for Vogue as a teenager, had a nervous breakdown after having her two children, left her writer husband and children for the French sculptor Jean Tinguely, and moved to an artists' colony where "anything went". She couldn't be bothered with any version of domestic bliss so they parted after a year (although they never divorced) and she pursued her own individual artistic path, which led her work into many wondrous places and spaces.

At once childlike and simultaneously ultra-adult, this is a singular show by an artist who reinvented her world and made others see their own worlds with fresh eyes.

Niki de Saint Phalle: The Eric & Jean Cass Gift, Gallery of Modern Art, Royal Exchange Square, Glasgow (0141 287 3050, www.glasgowlife.org.uk), until March 2014.