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The phrase "pivot back and forth" is correct and can be used in written English
This phrase is typically used to describe a person or object that is moving in a regular, back-and-forth motion, such as a pendulum. For example, "The patient's foot pivoted back and forth as the doctor examined it."
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Booyah raised more than $24.5 million from top-tier venture firms like Kleiner Perkins and Accel but it pivoted back and forth between Facebook and mobile platforms with varying success.
A red rudder pivots back and forth below the keel of ship, stirring the still air in the store, a prop shop called Jadis that first opened in Santa Monica in 1976.
His statement was the latest pivot in back-and-forth negotiations over a transfer of power, even as Mr. Saleh tries to frame the terms under which he would leave.
But the Florida senator quickly pivoted to his talking points, going on to repeat the same answer three times in a row in a brutal back-and-forth.
The back-and-forth continues for weeks.
There is back-and-forth on this.
It's a back-and-forth conversation.
Sometime in the third quarter of Game 5, he scurried into position, near the foul line, then pivoted blindly, back to the basket, back and forth, until, finally, he spun around and upward into a tilting fallaway.
"Still, you're navigating and pivoting all the time — it's like a football player carrying the ball back and forth," Gates said.
Back and forth.
They speculated back and forth.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com