This paper presents a new fundamental approach to modal participation analysis of linear time-invariant systems, leading to new insights and new formulas for modal participation factors. Modal participation factors were introduced over a quarter century ago as a way of measuring the relative participation of modes in states, and of states in modes, for linear time-invariant systems. Participation factors have proved their usefulness in the field of electric power systems and in other applications. However, in the current understanding, it is routinely taken for granted that the measure of participation of modes in states is identical to that for participation of states in modes. Here, a new analysis using averaging over an uncertain set of system initial conditions yields the conclusion that these quantities(participation of modes in states and participation of states in modes) should not be viewed as interchangeable. In fact, it is proposed that a new definition and calculation} replace the existing ones for state in mode participation factors, while the previously existing participation factors definition and formula should be retained but viewed only as mode in state participation factors. Examples are used throughout the paper to illustrate the issues addressed and results obtained.