  We define piecewise-linear and birational analogues of the toggle-involutions
on order ideals of posets studied by Striker and Williams and use them to
define corresponding analogues of rowmotion and promotion that share many of
the properties of combinatorial rowmotion and promotion. Piecewise-linear
rowmotion (like birational rowmotion) admits an alternative definition related
to Stanley's transfer map for the order polytope; piecewise-linear promotion
relates to Sch\"utzenberger promotion for semistandard Young tableaux. The
three settings for these dynamical systems (combinatorial, piecewise-linear,
and birational) are intimately related: the piecewise-linear operations arise
as tropicalizations of the birational operations, and the combinatorial
operations arise as restrictions of the piecewise-linear operations to the
vertex-set of the order polytope. In the case where the poset is of the form
$[a] \times [b]$, we exploit a reciprocal symmetry property recently proved by
Grinberg and Roby to show that birational rowmotion (and consequently
piecewise-linear rowmotion) is of order $a+b$. This yields a new proof of a
theorem of Cameron and Fon-der-Flaass. Our proofs make use of the
correspondence between rowmotion and promotion orbits discovered by Striker and
Williams, which we make more concrete. We also prove some homomesy results,
showing that for certain functions $f$, the average value of $f$ over each
rowmotion/promotion orbit is independent of the orbit chosen.
