  Facet-defining inequalities of the symmetric Traveling Salesman Problem (TSP)
polytope play a prominent role in both polyhedral TSP research and
state-of-the-art TSP solvers. In this paper, we introduce a new class of
facet-defining inequalities, the \emph{circlet inequalities}. These
inequalities were first conjectured in Gutekunst and Williamson \cite{Gut19b}
when studying Circulant TSP, and they provide a bridge between polyhedral TSP
research and number-theoretic investigations of Hamiltonian cycles stemming
from a conjecture due to Marco Buratti in 2017. The circlet inequalities
exhibit circulant symmetry by placing the same weight on all edges of a given
length; our main proof exploits this symmetry to prove the validity of the
circlet inequalities. We then show that the circlet inequalities are
facet-defining and compute their strength following Goemans \cite{Goe95b}; they
achieve the same worst-case strength as the similarly circulant crown
inequalities of Naddef and Rinaldi \cite{Nad92}, but are generally stronger.
