For input redundant systems, it is possible to construct distinct inputs producing identical outputs, from the same initial state. Thus, a system is input redundant if it is not left-invertible. In this paper, a rigorous framework is proposed to support this new definition. From the observation that state trajectories induced by the two inputs can be identical or not, an enriched taxonomy of input redundancy is also designed. A comprehensive set of characterizations is associated with the proposed definitions. A degree of redundancy is also conceived. This allows to carry out a thorough comparison with the state-of-the-art. Finally, the fact that over-actuated systems are input redundant is formally proved. This fact leads to a control design framework which can easily cope with over-actuated systems equipped with dynamical actuators.