It's Not A Magic Square

In how many distinct ways can I fill a \(3\times3\) grid with the integers 1 through 9 (each occurring exactly once) such that all cells in the grid are coprime with the neighbors? (Note that rotations and reflections count as distinct ways of filling the grid.)

Clarification: A cell's neighbors are the cells that share a full side with it. If two cells share a corner but not a side, they are not neighbors.

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