  We consider generalizations of parity polytopes whose variables, in addition
to a parity constraint, satisfy certain ordering constraints. More precisely,
the variable domain is partitioned into $k$ contiguous groups, and within each
group, we require that $x_i \geq x_{i+1}$ for all relevant $i$. Such
constraints are used to break symmetry after replacing an integer variable by a
sum of binary variables, so-called binarization. We provide extended
formulations for such polytopes, derive a complete outer description, and
present a separation algorithm for the new constraints. It turns out that
applying binarization and only enforcing parity constraints on the new
variables is often a bad idea. For our application, an integer programming
model for the graphic traveling salesman problem, we observe that parity
constraints do not improve the dual bounds, and we provide a theoretical
explanation of this effect.
