Summary: We consider uniform plane Hall plates with an arbitrary number of holes exposed to a uniform perpendicular magnetic field of arbitrary strength. The plates have extended contacts on the outer boundary and on the boundaries of the holes. No symmetry is presumed. Pairs of complementary Hall plates are defined, where contacts and insulating segments on the boundaries are swapped. A unique stream function exists in these multiply-connected domains if the total current through each boundary vanishes. Then the voltages between neighbouring contacts and the currents into contacts of pairs of complementary Hall plates are linked in a peculiar way, which results in identical power density. Relations between the impedances of complementary multiply-connected Hall plates are derived. The prominent role of complementary symmetric Hall plates – with or without holes – is revealed. For arbitrary magnetic field, their resistance equals the one of a square without holes and with contacts fully covering two opposite edges.