Representation theory of inner forms

I vote for "misguided". The representation theory of inner forms are certainly not the "same". What is true (over a local field) is they have the same L-group. A precise version is: the L-packets for G embed in the L-packets for the quasisplit form (assuming the local Langlands conjectures of course).

Well, it is not misguided if by "representation" you mean "algebraic representation". More generally, if $E$ is a right $G$-torsor over $Spec F$ and $X$ is a $G$-variety you can form a ``twisted form'' $E\wedge_G X=E\times X/(e,x)\sim (eg,gx)$ which is ${}_EG$-variety, where ${}_EG$ is the inner twisted form of $G$ corresponding to $E$. This gives an equivalence between the category of $G$-varieties and the category of ${}_EG$-varieties. The same construction applies to algebraic representations (and the equivalence is additive and preserves tensor products).