  In modular invariant models of flavor, observables must be modular invariant.
The observables discussed so far in the literature are functions of the modulus
$\tau$ and its conjugate, $\bar\tau$. We point out that certain combinations of
observables depend only on $\tau$, i.e. are meromorphic, and in some cases even
holomorphic functions of $\tau$. These functions, which we dub ``invariants''
in this Letter, are highly constrained, renormalization group invariant, and
allow us to derive many of the models' features without the need for extensive
parameter scans. We illustrate the robustness of these invariants in two
existing models in the literature based on modular symmetries, $\Gamma_{3}$ and
$\Gamma_{5}$. We find that, in some cases, the invariants give rise to robust
relations among physical observables that are independent of $\tau$.
Furthermore, there are instances where additional symmetries exist among the
invariants. These symmetries are relevant phenomenologically and may provide a
dynamical way to realize symmetries of mass matrices.

High Energy Physics - Phenomenology;High Energy Physics - Theory