In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.

Comment: 10 pages, 5 figures. To appear in Discrete Mathematics and Theoretical Computer Science, special issue for Permutation Patterns 2015