We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a certain poset and gyration on partial fully-packed loop configurations are in equivariant bijection.