Consider the canonical pointed symmetric sequence of simplicial sets $S$ defined such that $S_0=S^0$ and $S_n=S^n$. This is an ordered sequence of simplicial sets with a natural symmetric group action defined by permutation of the suspensions. By abstract nonsense, we can show that this category has a symmetric monoidal closed product. The object $S$ admits the natural structure of a monoid for this tensor product. The category of symmetric spectra becomes the category of modules over this monoid.
What's deep and intricate about this object? Well, I just read a paper by A. Salch that shows that the category of commutative $S$-algebras models the proposed theory of the field with one element!
The category of modules over this monoid is the category of symmetric spectra!