A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.