Let $\pi$ be the automorphic representation of ${\rm GL}_2$ or $B^\times$ ($B$ an indefinite quaternion algebra over $\Bbb Q$) of central character $\varepsilon$ attached to some holomorphic newform $f$ of nebentypus $\varepsilon$.
Given $\pi$ we can consider
the contragredient (or "dual") representation $\check\pi$ which is isomorphic to $\pi\otimes\varepsilon^{-1}$;
the "complex conjugate" representation $\overline\pi$ (thinking of $\pi$ as an irreducible component of $L^2(G_{\Bbb Q}\backslash G_{\Bbb A},\varepsilon)$ we can define $\overline\pi=\{\overline F\mid F\in\pi\}$).
Both $\check\pi$ and $\overline\pi$ have central character $\varepsilon^{-1}$ (because $\varepsilon$ is unitary) and both definitions are involutory. Is there a chance that $\check\pi\simeq\overline\pi$?
This could be checked at the level of local components. Given $\pi=\bigotimes_{p\leq\infty}\pi_p$, it is clear that $\check\pi=\bigotimes_{p\leq\infty}(\pi_p\otimes\varepsilon_p^{-1})$ but it is not obvious to me the effect of $\pi\mapsto\overline\pi$ on local components.
Clearly, by multiplicity one, an identity of local components could be checked just for unramified primes.