If you identify the Hecke algebra with the principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlands dual options you pick) such that $T_y$ is the class of a Verma module with highest weight $-y\rho-\rho$ (so $T_1$ corresponds to the anti-dominant Verma), then the $C'$-basis will match with the classes of tilting modules, and the $C$ basis with the simple modules (I tend to prefer matching the $C'$-basis with projectives, but then the $C$-basis is something horrible).

I'm not really sure where this is written down properly; philosophically, the point is that everything will be fixed in place once you decide what functor to send the bar involution to. If you send it the contragredient dual, then simples and tiltings are obvious sets of modules fixed by this duality, and they satisfy the triangularities you've written above.

If you identify the Hecke algebra with the Grothendieck principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlands dual options you pick doesn't matter) such that $T_y$ is the class of a Verma module with highest weight $-y\rho-\rho$ (so $T_1$ corresponds to the anti-dominant Verma), then the $C'$-basis will match with the classes of tilting modules, and the $C$ basis with the simple modules (I tend to prefer matching the $C'$-basis with projectives, but then the $C$-basis is something horrible).

I'm not really sure where this is written down properly; philosophically, the point is that everything will be fixed in place once you decide what functor to send the bar involution to. If you send it the contragredient dual, then simples and tiltings are obvious sets of modules fixed by this duality, and they satisfy the triangularities you've written above.