I bet the washer is twisting-and-moving in the same direction that the washer's agitator spins, especially when it's spinning out the water. If not at that time, it may be doing it like mine. My washer agitates hard to the left and not-so-hard to the right. It's like a jerking motion but more to one side than the other.
As Joe C. suggests, you need to prevent the unit from sliding on your slick, new floor. You could look into getting the same type of 'rubber feet' that your grandparents use to put under the feet of their furniture so it wouldn't scratch the hardwood floor. They're usually located in the hardware section of the Home Center. My best to ya and we live and learn. :)