  We introduce the notions of symmetric and symmetrizable representations of
$\text{SL}_2(\mathbb{Z})$. The linear representations of
$\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric
and have congruence kernel. Conversely, one may also reconstruct modular data
from finite-dimensional symmetric, congruence representations of
$\text{SL}_2(\mathbb{Z})$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry
of some Weil representations at prime power levels, we prove that all
finite-dimensional congruence representations of $\text{SL}_2(\mathbb{Z})$ are
symmetrizable. We also provide examples of unsymmetrizable noncongruence
representations of $\text{SL}_2(\mathbb{Z})$ that are subrepresentations of a
symmetric one.
