Summary: A fence is a poset with elements \(F=\{x_1,x_2,\ldots,x_n\}\) and covers \[ x_1 \triangleleft x_2 \triangleleft \cdots \triangleleft x_a \triangleright x_{a+1} \triangleright \cdots \triangleright x_b \triangleleft x_{b+1} \triangleleft \cdots, \] where \(a,b,\ldots\) are positive integers. We investigate rowmotion on antichains and ideals of \(F\). In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a homomesy result for all self-dual posets and show that any two Coxeter elements in certain toggle groups behave similarly with respect to homomesies which are linear combinations of ideal indicator functions. We end with some conjectures and avenues for future research.