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The Symmetric Traveling Salesman Polytope $S_n$ for a fixed number $n$ ofcities is a face of the corresponding Graphical Traveling Salesman Polyhedron$P_n$. This has been used to study facets of $S_n$ using $P_n$ as a tool. Inthis paper, we study the operation of "rotating" (or "lifting") validinequalities for $S_n$ to obtain a valid inequalities for $P_n$.  As an application, we describe a surprising relationship between (a) theparsimonious property of relaxations of the Symmetric Traveling SalesmanPolytope and (b) a connectivity property of the ridge graph of the GraphicalTraveling Salesman Polyhedron.

On the facial structure of Symmetric and Graphical Traveling Salesman Polyhedra