Shifted twisted Yangians of quasi-split ADE types

Associated to all quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, we introduce the shifted iYangians ${}^\imath Y_\mu$ and establish their PBW bases. We construct the iGKLO representations of ${}^\imath Y_\mu$, which factor through quotients called truncated shifted iYangians ${}^\imath Y_\mu^\lambda$. In type AI with $\mu$ dominant, a variant of ${}^\imath Y_\mu^{N\varpi_1^\vee}$ is identified with the truncated shifted iYangians in another definition, which are isomorphic to finite W-algebras of type BCD. These new family of algebras has connections and applications to fixed point loci of affine Grassmannian slices which will be developed in a sequel.