A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initial- boundary value problems for linear, constant-coefficient evolution equations of arbitrary order on a finite domain. We use Fokas' method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify sufficient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a well-posed problem. We also discuss the spectral theory of the associated linear two-point ordinary differential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initial-boundary value problems.