To convert a direction to an angle represented with a value range of -Pi and Pi where the 0 degree mark is Vector2.Up and then converting that value from degrees back to a heading direction.

DirToAngleAndBack.cs

public Vector2 direction;

[ContextMenu("Test dir to angle and back again")]
void DirToAngle()
{
	var angleOfDir = Vector3.Angle(direction, transform.up);
	var cw = Vector3.Cross(transform.up, direction).z < 0f;
	if (!cw)
	{
		angleOfDir = -angleOfDir;
	}

	Debug.Log($"Angle is {angleOfDir}");
    
	var backToDir = Quaternion.AngleAxis(-angleOfDir, Vector3.forward) * Vector3.up;

	Debug.Log($"Converted back to dir: {(Vector2)backToDir}");
}