Forbidden rectangles in compacta

We establish negative results about "rectangular" local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type \omega x \omega_2. For another, the compactum \beta\omega has no nontrivially rectangular local bases, and the same is consistently true of \beta\omega \ \omega: no local base in \beta\omega has cofinal type \kappa x c if \kappa < m_{\sigma-n-linked} for some n in [1,\omega). Also, CH implies that every local base in \beta\omega \ \omega has the same cofinal type as one in \beta\omega. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on \omega always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on \omega is commutative modulo cofinal equivalence.