We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset P as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When P=[a]×[b], a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order a+b. We prove some homomesy results, showing that for certain functions f, the average of f over each rowmotion/promotion orbit is independent of the orbit chosen.