We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph G. These generalize the well-known tractable planar case, and they include the genus of G, its apex number (the minimum number of vertices whose removal renders G planar), and its Hadwiger number (the size of a largest clique minor).To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain non-existing gadgets F as linear combinations of t = O(1) existing gadgets. If a graph G features k occurrences of F , we can then reduce G to t k graphs that feature only existing gadgets, thus enabling parameterized reductions.As applications of this technique, we simplify known 4 g n O(1) time algorithms for PerfMatch on graphs of genus g. Orthogonally to this, we show #W[1]-hardness of the permanent on k-apex graphs, implying its #W[1]-hardness under the Hadwiger number. Additionally, we rule out n o(k/ log k) time algorithms under the counting exponential-time hypothesis #ETH.Finally, we use combined matchgates to prove ⊕W[1]-hardness of evaluating the permanent modulo 2 k , complementing an O(n 4k−3 ) time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of n Ω(k/ log k) under the parity version ⊕ETH of the exponential-time hypothesis.