A noncommutative polynomial (an element of the free algebra) can be viewed as a function from tuples of square matrices to square matrices, of arbitrary sizes. This view is adopted in free analysis, where noncommutative polynomials are prototypical examples of more general noncommutative functions. From this perspective, there is a common problem of how much info about a noncommutative polynomial can be recovered from the features of its matrix evaluations. This talk addresses an aspect of this problem, by classifying pairs of noncommutative polynomials whose evaluations share common spectral properties. Concretely, the talk considers pairs of noncommutative polynomials that have pointwise the same rank, eigenvalues, norm, or similar values. For each of these pointwise equivalences, a counterpart in the free algebra (that is, a structure governing it) is presented. Various examples and further directions are also discussed.