  To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a
matroid structure called {\em the bouquet graph} of $A$ and introduce another
toric ideal called {\em the bouquet ideal} of $A$. We show how these objects
capture the essential combinatorial and algebraic information about $I_A$.
Passing from the toric ideal to its bouquet ideal reveals a structure that
allows us to classify several cases. For example, on the one end of the
spectrum, there are ideals that we call {\em stable}, for which bouquets
capture the complexity of various generating sets as well as the minimal free
resolution. On the other end of the spectrum lie toric ideals whose various
bases (e.g., minimal generating sets, Gr\"obner, Graver bases) coincide. Apart
from allowing for classification-type results, bouquets provide a new way to
construct families of examples of toric ideals with various interesting
properties, such as robustness, genericity, and unimodularity. The new bouquet
framework can be used to provide a characterization of toric ideals whose
Graver basis, the universal Gr\"obner basis, any reduced Gr\"obner basis and
any minimal generating set coincide.
