  A truncated Toeplitz operator is the compression $A_{\phi}:\K_{\Theta} \to
\K_{\Theta}$ of a Toeplitz operator $T_{\phi}:H^2\to H^2$ to a model space
$\K_{\Theta} := H^2 \ominus \Theta H^2$. For $\Theta$ inner, let $\T_{\Theta}$
denote the set of all bounded truncated Toeplitz operators on $\K_{\Theta}$.
Our main result is a necessary and sufficient condition on inner functions
$\Theta_1$ and $\Theta_2$ which guarantees that $\mathcal{T}_{\Theta_1}$ and
$\mathcal{T}_{\Theta_2}$ are spatially isomorphic (i.e., $U\T_{\Theta_1} =
\T_{\Theta_2}U$ for some unitary $U:\K_{\Theta_1} \to \K_{\Theta_2}$). We also
study operators which are unitarily equivalent to truncated Toeplitz operators
and we prove that every operator on a finite dimensional Hilbert space is
similar to a truncated Toeplitz operator.
