The Symmetric Traveling Salesman Polytope Sn for a fixed number n of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron Pn. This has been used to study facets of Sn
Nearby in the stack
using
Pn
as a tool. In this paper, we study the operation of "rotating" (or "lifting") valid inequalities for
Sn
to obtain a valid inequalities for
Pn
. As an application, we describe a surprising relationship between (a) the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope and (b) a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron.