What is the opposite of "counter-rotating"?

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corotate, intransitive verb: to rotate in conjunction with or at the same rate as another rotating body

Merriam Websters

Also see

Corotational: That rotates in conjunction with another body

Wikipedia

Oxford Reference gives the example of two bodies spinning about their common centre of mass, and therefore necessarily in the same clockwise or anti-clockwise way.

Where one body orbits another, the circumstance in which one or both bodies have an axial rotation period the same as the orbiting body's revolution period. The co-rotating body therefore keeps one face permanently turned towards the other, as in the case of the Moon facing the Earth. In the case of Pluto and its satellite Charon, both bodies are in co-rotation and so each keeps the same face turned to the other. See also spin–orbit coupling; synchronous rotation.

Oxford Reference