The second sentence is incorrect. The symmetric square $L$-function of a Maass form $f$ (on the upper half-plane) has a pole at $s=1$ if and only if $f$ is dihedral. On the other hand, most self-dual Maass forms are not dihedral. For example, the Maass forms of level $1$ are self-dual but not dihedral.
In fact if a Maass form $f$ (on the upper half-plane) is dihedral, i.e. $L(s,f)=L(s,\eta)$ as in the original post, then $f$ is self-dual if and only if $\eta$ is a quadratic Hecke character.