We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: – properties of truncated Toeplitz operators; – Carleson-type embedding theorems for the coinvariant subspaces; – factorizations of pseudocontinuable functions from $H^1$. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.