Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation)
of the Langlands dual group $^LG$.  

The irreducible representations of $G$ have additional structure related to tensoring representations:  Given representations $\alpha, \beta, \gamma$ of $G$,
we have the invariant subspace $V_{\alpha\beta\gamma}$ of the tensor product $\alpha\otimes\beta\otimes\gamma$.
(Or, if you prefer, the space $V_{\alpha\beta}^{\gamma^*}$ of homomorphisms from $\gamma^*$ tp $\alpha\otimes\beta$.)

> Is there an intrinsic way to define a Langlands dual structure on the
> cocharacters of $^LG$?  In other words, in a natural way (and without
> using Langlands duality) associate to cocharacters $a,b,c$ of $^LG$ a
> vector space $V_{abc}$?

One possibility would be to think of the cocharacters as (equivalence classes of) geodesics in $^LG$ and replace the tensoring of $G$ representations with the splicing of $^LG$ geodesics.  The resulting families of broken geodesics $a \cdot b$ could flow to actual geodesics $\{c_i\}$.  Before pursuing this idea I wanted to check whether there are already known answers to the main question above.