At present, the most successful approach for solving large-scale instances of the Symmetric Traveling Salesman Problem to optimality is branch-and-cut. The success of branch-and-cut is due in large part to the availability of effective separation procedures; that is, routines for identifying violated linear constraints. For two particular classes of constraints, known as comb and domino-parity constraints, it has been shown that separation becomes easier when the underlying graph is planar. We continue this line of research by showing how to exploit planarity in the separation of three other classes of constraints: subtour elimination, 2-matching and simple domino-parity constraints.