Although there are some who dismiss his works as doodles done at the telephone, there are enough collectors and curators who think otherwise.
One historian has likened Twombly to Leonardo in the way that drawn lines, 'like a natural phenomenon, unfold in space and time'.
Eli Broad, a Californian collector, was the man intent on buying Sotheby's example. The untitled work was of 14 such paintings dating from 1969.
According to Sotheby's catalogue, this work has 'thin, arching rectangles, elongated phallic forms, cylinders and heart shapes that cascade down from the upper right corner of the canvas toward the left edge along the horizontal centre . . . a charting of the labyrinth of the existence'.
Twombly himself said in 1957: 'Every line is thus the actual experience with its unique story. It does not illustrate; it is the perception of its own realisation.'
Strangely enough though, considering his following, the previous night Christie's had failed to sell a Twombly blackboard painting for the expected dollars 2.2m.
That was the only blip in otherwise buoyant contemporary art sales at both auction- houses. Some 79 per cent of Sotheby's auction found buyers, compared with 81 per cent at Christie's.
Although Sotheby's did not have prices at the level of a pounds 3.85m Gorky abstract achieved by Christie's, it did break the record for a sculpture by Alexander Calder, the American artist who was among the first to introduce real movement into sculptural art.
An American collector bought Constellation, his 1960 painted metal standing mobile, for dollars 1.81m ( pounds 1.22m), against an estimate of just dollars 500,000 to dollars 700,000.
David Hockney's first view of California and swimming- pool picture, California Art Collector, 1964, sold for dollars 1.02m ( pounds 687,100); its top estimate was dollars 800,000.
Among works by younger artists, New Double Shelton Wet Dry - a pair of vacuum cleaners under a plexiglass cabinet - by Jeff Koons, master of high kitsch, sold for dollars 112,500.
When staff at Sotheby's in London announced last month the discovery of two Constable watercolours, they could allow themselves to feel confident of getting good prices.
And they did, when the watercolours came to auction yesterday. At least four bidders were competing for each one.
Both works went well above estimates of around pounds 20,000: A Ploughman near East Bergholt Overlooking the Stour Estuary and Dedham Vale from the Lane between East Bergholt and Flatford sold for pounds 95,000 and pounds 54,300 respectively.
They were bought by an anonymous private British collector, bidding over the telephone, who intends to keep them in this country.