  Asymmetric dual truncated Toeplitz operators acting between the orthogonal
complements of two (eventually different) model spaces are introduced and
studied. They are shown to be equivalent after extension to paired operators on
$L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in
$L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the
inverse symbol. Relations with Carleson's corona theorem are also established.
These results are used to study the Fredholmness, the invertibility and the
spectra of various classes of dual truncated Toeplitz operators.
