A balanced seesaw is simply one that isn’t experiencing any torque—the net torque on it is zero. Because there is no torque on it, it isn’t undergoing any angular acceleration and its angular velocity is constant. If it happens to be horizontal and motionless, then it will stay that way. But it could also be tilted or even rotating at a steady rate.
Angular speed is the measure of how quickly an object is turning. For example, an object that is spinning once each second has an angular speed of “1 rotation-per-second,” or equivalently “360 degrees-per-second.” Angular velocity is a combination of angular speed and the direction of the rotation. For example, a clock lying on its back and facing upward has a minute hand with an angular velocity of “1 rotation-per-hour in the downward direction.” The downward direction reflects the fact that the minute hand pivots about a vertical axis and that your right hand thumb would point downward if you were to curl your fingers in the direction of the minute hand’s rotation.
The rule that’s used in the mechanics of rotation is always the right hand rule and that’s important. It represents a choice made long ago about how to describe an object’s rotation. Having made that choice, it says that the minute hand of a clock (which naturally rotates clockwise) points into the clock. You know that because if you curl the fingers of your right hand in the direction that the minute hand is turning, your extended thumb will point into the clock. There is no left hand rule because that was not the choice made long ago.