A very strange ballet, but less strange if you recall that Balanchine -- like many New Yorkers -- had seen gagaku performances in 1959. Gagaku and other forms of Japanese classical performance were introduced to NYC in the 50s and early 60s and made a great impression.
Here is Jack Anderson, NY Times, writing in 1996. He is referring to Lincoln Kirstein, Balanchine's partner in the creation of the New York City Ballet.:
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An admirer of the traditional arts of Japan, Mr. Kirstein helped arrange a 1959 American tour for Gagaku, an all-male troupe that is the Japanese Imperial Household Musicians and Dancers. Gagaku has rarely performed outside the Imperial Music Pavilion in Tokyo and some of the great Japanese shrines.
Balanchine's take on all this focuses on a mating ritual. My memories of the original cast are limited to on (a) Allegra Kent, (b) the costumes and design, and © Allegra Kent.

For some reason, I can't recall Villella at all. I DO recall Arthur Mitchell later on. There are elements in the work between the man and woman that remind me of Agon pdd.
I tracked down Anna Kisselgoff's review of a 1981 revival by NYCB. She goes into the background a bit.
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"Bugaku" was inspired by the Japanese emperor's highly refined ensemble of musicians and dancers known as Gagaku, and by its music, Bugaku. But the choreography's high point is an erotic duet in Balanchine's stylized grand manner. Ballet lore has it that a Japanese delegation invited to the 1963 premiere was not amused. Imagine a Japanese ballet about sex in the Oval Office at the White House. Whatever possible interpretation might result from the content, the ballet is blameless in its form. Take away Karinska's Japanese wigs, flower-petal tutus and mock kimonos and the choreography will reveal its kinship with Balanchine's most experimental works: angular and convoluted, usually danced in leotards. As a rule these are set to modern music, and "Bugaku" has an ingenious commissioned score by a major Japanese contemporary composer, Toshiro Mayuzumi.
Heather Watts and Jock Soto were the geisha and samurai escorted by an ensemble of male and female attendants. Miss Watts has honed the ballerina role to inventive perfection. She can scurry on toe with the modesty of her retinue, 75 percent of which is named Jennifer in one spelling or another: Jennifer Fuchs, Jenifer Ringer, Jennifer Tinsley and Rita Norona. But the assertiveness of her jumps onto toe, feet held parallel, presages her boldness in the famous pas de deux.
Stretched to the extreme as she begins in a low-slung position mirrored by Mr. Soto, she moves her leg eventually into a split up her partner's chest. The most acrobatic of embraces is followed by a ceremonial dance for all, full of floating veils kicked up by flicks of a leg. Mr. Soto's performance is especially likable: light in his leaps and beats but solid as a masculine foil, he exudes grace without exaggeration.
So far, I've tended to think about the MCB revival mainly as an exercise in ballet history. Now, thanks to this thread, I'm hoping to enjoy it on a higher and deeper level than I was capable of in my youth.
Looking forward to hearing what others have to say, especially those who have seen it in NYCB revivals over the years.