  For a graph $G$, its Tutte symmetric function $XB_G$ generalizes both the
Tutte polynomial $T_G$ and the chromatic symmetric function $X_G$. We may also
consider $XB$ as a map from the $t$-extended Hopf algebra $\mathbb{G}[t]$ of
labelled graphs to symmetric functions.
  We show that the kernel of $XB$ is generated by vertex-relabellings and a
finite set of modular relations, in the same style as a recent analogous result
by Penaguiao on the chromatic symmetric function $X$. In particular, we find
one such relation that generalizes the well-known triangular modular relation
of Orellana and Scott, and build upon this to give a modular relation of the
Tutte symmetric function for any two-edge-connected graph that generalizes the
$n$-cycle relation of Dahlberg and van Willigenburg. Additionally, we give a
structural characterization of all local modular relations of the chromatic and
Tutte symmetric functions, and prove that there is no single local modification
that preserves either function on simple graphs.
  We also give an expansion relating $XB_G$ to $X_{G/S}$ as $S$ ranges over all
subsets of $E(G)$, use this to extend results on the chromatic symmetric
function to the Tutte symmetric function, and show that analogous formulas hold
for a Tutte quasisymmetric function on digraphs.
