  Using the non-semisimple Temperley-Lieb calculus, we study the additive and
monoidal structure of the category of tilting modules for $\mathrm{SL}_{2}$ in
the mixed case. This simultaneously generalizes the semisimple situation, the
case of the complex quantum group at a root of unity, and the algebraic group
case in positive characteristic. We describe character formulas and give a
presentation of the category of tilting modules as an additive category via a
quiver with relations. Turning to the monoidal structure, we describe fusion
rules and obtain an explicit recursive description of the appropriate analog of
Jones-Wenzl projectors. We also discuss certain theta values, the tensor
ideals, mixed Verlinde quotients and the non-degeneracy of the braiding.
