In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermite equivalence class of a polynomial has a very natural interpretation in terms of the invariant ring and invariant ideal associated with the polynomial. We apply this interpretation to shed light on the relationship between Hermite equivalence and more familiar notions of polynomial equivalence, such as ${\rm GL}_2(\mathbb{Z})$- and $\mathbb{Z}$-equivalence. Specifically, we prove that ${\rm GL}_2(\mathbb{Z})$-equivalent polynomials are Hermite equivalent and, for polynomials of degree $2$ or $3$, the converse is also true. On the other hand, for every $n\geq 4$, we give infinite collections of examples of polynomials $f,g\in \mathbb{Z}[X]$ of degree $n$ that are Hermite equivalent but not ${\rm GL}_2(\mathbb{Z})$-equivalent.

Compared with the previous version we have inserted some changes and corrections suggested by the anonymous referee. This is the final version. It will appear in a special volume of Acta Arithmetica to the memory of Professor Andrzej Schinzel