  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.
