  We introduce several classes of polytopes contained in $[0,1]^n$ and cut out
by inequalities involving sums of consecutive coordinates. We show that the
normalized volumes of these polytopes enumerate circular extensions of certain
partial cyclic orders. Among other things this gives a new point of view on a
question popularized by Stanley. We also provide a combinatorial interpretation
of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of
descents of total cyclic orders. The Euler numbers, the Eulerian numbers and
the Narayana numbers appear as special cases.
