Does the inverse of gravity exists? I am not talking about an antigravity effect that would cancel gravity, rather if there is a repulsive force, like when you use two magnets and flip one of them, where it becomes obvious that one is pushing away the other.

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No.

No such effect has been discovered. Gravity seems to be always attractive, never repulsive. Nor has a particle with negative mass ever been discovered. In particular, antimatter appears to have positive mass.

It is not theoretically impossible, but a particle with negative mass would be very odd, as the faster it moved, the less momentum and energy it would have.