  The first author recently introduced toric promotion, an operator that acts
on the labelings of a graph $G$ and serves as a cyclic analogue of
Sch\"utzenberger's promotion operator. Toric promotion is defined as the
composition of certain toggle operators, listed in a natural cyclic order. We
consider more general permutoric promotion operators, which are defined as
compositions of the same toggles, but in permuted orders. We settle a
conjecture of the first author by determining the orders of all permutoric
promotion operators when $G$ is a path graph. In fact, we completely
characterize the orbit structures of these operators, showing that they satisfy
the cyclic sieving phenomenon. The first half of our proof requires us to
introduce and analyze new broken promotion operators, which can be interpreted
via globs of liquid gliding on a path graph. For the latter half of our proof,
we reformulate the dynamics of permutoric promotion via stones sliding along a
cycle graph and coins colliding with each other on a path graph.
