New York ? For so many American kids, it’s their first — and sometimes only — exposure to ballet: A holiday performance of “The Nutcracker,” that classic tale of a little girl’s dreamlike encounters with battling mice and toy soldiers, a dashing prince, a growing Christmas tree and a land of sweets.

There’s one in virtually every town. But until now, if you lived in Anchorage or Omaha or Tulsa, you weren’t able to see what many consider the Nutcracker gold standard: the late George Balanchine’s classic production for New York City Ballet.

On Tuesday, though, the company will beam a live “Nutcracker” performance in high-definition from Lincoln Center to some 560 movie theaters in all 50 states, followed the next night by a live broadcast on public television.

In taking its “Nutcracker” national, the company hopes to promote its brand, earn new revenue and join a growing trend of HD transmissions of live performance, pioneered by the Metropolitan Opera.

“There’s nothing like the real thing,” says an admittedly “totally biased” Peter Martins, the company’s ballet master in chief. “Of course, the hope is that if this works — however you define success — it will become a new way to show the world what we do so well.”

Martins — a former principal dancer with New York City Ballet under his legendary predecessor, Balanchine — grew up in Denmark and thus wasn’t exposed to “The Nutcracker” until he came to New York as a young adult. But he’s well aware of its crucial role as an introduction to ballet for generations of Americans — including virtually all his own dancers.

One of them is Ashley Bouder, who was 6 when she saw her first “Nutcracker” in Carlisle, Pa. — as it happened, she also performed in it as an angel. It launched a lifetime of dancing, and Bouder, who turns 28 today, is now an NYCB principal, known for her fearless, attacking style in jumps and turns.

“I do think this particular ‘Nutcracker’ is really special,” said Bouder, who will dance the brief but demanding role of Dewdrop in both the live performances. “Balanchine’s choreography is so wonderful, for children as well as adults.”

Indeed, the choreographer is known to have been particularly masterful with children — as in the scene where eight rosy-cheeked kids suddenly emerge from Mother Ginger’s 9-foot-wide skirt to perform an intricate yet buoyant dance.

Growing up in Spain, Joaquin De Luz never got to see a live “Nutcracker” as a child. The first “Nutcracker” he saw was Balanchine’s — on DVD, the 1993 version put out by New York City Ballet with the young Macaulay Culkin as the prince.

“It’s just magical,” said De Luz, who’ll play the main male role, the Cavalier, next week. “That growing tree in the first act just breaks your heart. This is the quintessential ‘Nutcracker’ — not too much of anything, not too little.”

De Luz will be dancing with his frequent partner, Megan Fairchild, as the Sugarplum Fairy. It’s better not to think too much about how many people will be watching, he said.

“This is a big deal,” De Luz said. “I’m just hoping they won’t do the close-ups when I’m wetting my lips or doing something else I shouldn’t be doing.”

That would be a minor mishap compared to last weekend, when Fairchild’s costume suddenly got caught on De Luz’s buttons during a lift. Such is the joy of live performing; he spent crucial seconds trying to yank the costumes apart before anyone got hurt.

“I finally ripped the whole thing,” he said. “I’m glad it happened last Saturday. It’s good to have an accident under your belt.”

It’s that coolness under pressure that Martins was looking for when casting the leads for next week’s performances. “I have 12 casts and they’re all fantastic,” Martins said. “But this particular cast is very, very reliable. They will always deliver.”

The ballet transmissions, produced by Live From Lincoln Center for the NCM Fathom theater network, are the first by an American ballet company (the Bolshoi Ballet and the Royal Ballet in London have done their own, as well as several opera companies and the National Theater in London.)