  We relate multiband truncated 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 multiband truncated shift, which turns out to be
considerably more complex than the truncated shift on model spaces, revealing
new and nontrivial connections with the boundary behaviour of the associated
inner function.
