The phrase never the twain shall meet means: two different people or things are totally incompatible.

(Twain is an archaic word for two—cf. mark twain.)

The phrase never the twain shall meet occurs, for example, in Managing drops in a bucket, by Duncan Adams, published in the Missoulian (Missoula, Montana, USA) of Thursday 2^nd March 2023:

The hard-bitten, blustering cattle rancher, he or she of pioneer progeny, sometimes occupies one extreme pole of the water allocation debate, and the fish-hugging, cattle-loathing, synthetic clad environmentalist stakes claim to the other.
And never the twain shall meet.

The phrase never the twain shall meet alludes to the following from Ballad of East and West, in Barrack-Room Ballads and Other Verses (London: Methuen and Co., 1892), by the English poet, short-story writer and novelist Joseph Rudyard Kipling (1865-1936):

Oh, East is East and West is West, and never the twain shall meet,
Till Earth and Sky stand presently at God’s great Judgment Seat.

The earliest occurrences that I have found of the phrase never the twain shall meet—not immediately preceded by East is East and West is West, and used without explicit reference to Rudyard Kipling—are as follows, in chronological order:

1-: From the review of Tess of the d’Urbervilles ¹, a play produced at the California Theatre, San Francisco, with the U.S. actress Minnie Maddern Fiske (born Marie Augusta Davey – 1865-1932) as Tess—review published The Examiner (San Francisco, California, USA) of Tuesday 5^th February 1901:

A man’s a man and a woman’s a woman, and “never the twain shall meet” on moral grounds.

¹ This play was based on Tess of the d’Urbervilles: A Pure Woman (1891), by the British novelist and poet Thomas Hardy (1840-1928).

2-: From The Morning Leader (London, England) of Thursday 7^th March 1901—the allusion to “the prophet of the Himalayas” is obscure:

SYRIA AND THE GRAPHOPHONE.
A FAMOUS PROPHECY THAT IS NOW RATHER AT A DISCOUNT.

“Never the twain shall meet,” says the prophet of the Himalayas; but it certainly seems as though his prophecy was at a discount, since east and west have met in the harem of the Sultan of Turkey.
For the graphophone from the west has travelled east, from America, via Lendon, to Turkey, where the veiled beauties of the seraglio may listen to the latest music-hall song from London, or the latest wild wheeze—à la Americaine—from Paris.

3-: From the review of A Japanese Nightingale (1901), by Onoto Watanna, pen-name of the Canadian author Winnifred Eaton (1875-1954)—review published in The Washington Times (Washington, District of Columbia, USA) of Sunday 20^th October 1901:

In reading it one feels the mysterious fascination of the land of cherry blossoms and chrysanthemums, and gets bewitching glimpses into the unknown world of native Japan. That the Japanese character possesses much that is alluring and beautiful, Lafcadio Hearn ² and other foreigners who know the land thoroughly have assured us. But the gulf between the East and the West is so wide and deep that “never the twain shall meet” except in some such half-satisfactory way as this.

² Lafcadio Hearn (1850-1904), also called Koizumi Yakumo, was an Irish-Greek-Japanese author, translator and teacher who introduced the culture and literature of Japan to the West.

4-: From Our London Correspondence, published in The Manchester Guardian (Manchester, Lancashire, England) of Monday 24^th November 1902:

“East London is east and West London is west, and never the twain shall meet.” In various shapes this belief has so long been proclaimed a truism that, like other dogmas, it has reached the inevitable stage of decadence and is believed only by courtesy.

5-: From A Plague of Pigeons, published in the Boston Evening Transcript (Boston, Massachusetts, USA) of Wednesday 6^th July 1904—reprinted from the London Chronicle:

The pigeons of the cathedral [i.e., St. Paul’s] at the present day offer an interesting study to the naturalist. There are two distinct colonies of them—the one on the east and the other on the west, “and never the twain shall meet.” As far as observation goes they are as well defined and separate as the two Houses of Convocation, and if a member of one were to stray into the other he would meet with a very unchristian reception.

6-: From a comment on the results of the municipal elections at King’s Lynn, published in the Lynn News & County Press for Norfolk, Isle of Ely, and South Lincolnshire (King’s Lynn, Norfolk, England) of Saturday 5^th November 1904:

Astounding is the fact that 25 people actually voted for Brown and Bunnett. That Labour men can see any municipal resemblance between these two candidates passes comprehension. One can understand the split vote of 38 between Bunnett and Green, but Brown is west and Bunnett is east, and never the twain shall meet!

7-: From the column Gossip of the Hour, published in the Herne Bay Press (Herne Bay, Kent, England) of Saturday 13^th January 1906:

A correspondent, one of our Liberal readers, writes:
Your note on Mr. Akers Douglas’ votes would be hard to beat, and you may well say that it will be considered subtle reasoning. It must stand as an ingenious display of sophistical argument that must have delighted your Conservative friends and caused your Liberal friends to gnash their teeth. Mr. Akers Douglas’ votes must be right or wrong. You have tried very hard to prove them right, and after reading your article it seems delightfully easy to prove that wrong is right, and right is wrong; but to me right is right and wrong is wrong, and “never the twain shall meet.”

8-: From the transcript of a speech that Randolph Rust (1854-1930) delivered on Wednesday 10^th January 1906 at the Municipal Committee, published in The Mirror (Port of Spain, Trinidad, Trinidad & Tobago) of Friday 26^th January 1906:

We are now asked to decide whether the Central Authority we have already stated in our opinion should be provided, should be composed partly of nominated and partly of elected members—that kind of authority which is known as a “mixed municipality.” […] I think Mr. Joseph Chamberlain must be credited with having originated the idea that a “mixed municipality” might be good for us in Port-of-Spain. The question then arises: where did Mr. Chamberlain get such a knowledge of mixed municipalities? We know he was a very busy man who occasionally had to take ideas from others, and it appeared Mr. Chamberlain did not get this idea from his own office—the Colonial Office—at all, but that it arose from the India Office. […] But, gentlemen, east is east and west is west—(a voice sotto voce: “And never the twain shall meet”)—and the two conditions are very different.

9-: From The Pall Mall Gazette (London, England) of Tuesday 27^th March 1906:

“Never the twain shall meet.” With Japanese bluejackets at Nelson’s tomb, however, and a Chinese Duke—the first Celestial we have known with that title—at the head of a mission to study constitutionalism in these islands, East and West seem in a fairer way of coming together than once seemed probable. To be sure, a Chinese Minister gave an evening party in London nearly thirty years ago—the first function of the kind that ever took place under Celestial auspices; and more than ten years earlier, a Chinese newspaper, “The Messenger of the Flying Dragon,” was published in this metropolis. A Chinese Ambassador has studied our commercial method of procedure. But the coming of a Celestial Duke to assimilate our constitutional method suggests a much longer stride Westwards.

10-: From the review of A Vision of India (1906), by the British journalist, historian and essayist Sidney Low (1857-1932)—review published in The Evening Citizen (Ottawa, Ontario, Canada) of Saturday 16^th June 1906:

Mr. Low has much that is important to say of the relation between the governing race and the governed. He notes how, despite education, breeding and riches, the native is never or hardly ever accepted as an equal by the Englishman, and though this is a very obvious difficulty in the governing of the empire and an increasing source of irritation, there seems no prospect of its ever coming to an end. “East is East and West is West,” and though mutual respect and mutual admiration may exist in every vital sense, “never the twain shall meet.”

11-: From Brains are essential in piano playing, published in the Oakland Tribune (Oakland, California, USA) of Sunday 7^th October 1906:

Personality in musical interpretation is style, and style is the man.
Ternina’s Tosca is Ternina ³ and Eames’s Tosca is Eames ⁴, and never the twain shall meet till earth and sky stand presently at God’s great judgment seat.

³ Milka Ternina (born Katarina Milka Trnina – 1863-1941) was a Croatian operatic soprano.
⁴ Emma Eames (1865-1952) was a U.S. operatic soprano.

12-: From the review of a play titled Strongheart, published in The World (Vancouver, British Columbia, Canada) of Monday 19^th November 1906:

Love is love, but duty is duty. Their sign posts point different ways and the curtain descends upon a lone Indian, whose hands are stretched to heaven, not so much in appeal, as in heart-rending submission—submission which still has in it more than a remnant of fighting power. It is the inevitable. White is white and red is red, and never the twain shall meet if each is true to their race.