  We relate dual-band general Toeplitz operators to block truncated Toeplitz
operators and, via equivalence after extension, with Toeplitz operators with $4
\times 4$ matrix symbols. We discuss their norm, their kernel, Fredhomlness,
invertibility and spectral properties in various situations, focusing on the
spectral properties of the dual-band shift, which turns out to be considerably
complex, leading to new and nontrivial connections with the boundary behaviour
of the associated inner function.
