  We prove that if $\phi$ is an analytic function bounded by 1 on the bidisk
and $\tau$ is a point in a face of the bidisk at which $\phi$ satisfies
Caratheodory's condition then both $\phi$ and the angular gradient $\nabla\phi$
exist and are constant on the face. Moreover, the class of all $\phi$ with
prescribed $\phi(\tau)$ and $\nabla\phi(\tau)$ can be parametrized in terms of
a function in the two-variable Pick class. As an application we solve an
interpolation problem with nodes that lie on faces of the bidisk.
