Finite semisymmetric quasigroups are in bijection with certain mappings between abstract polyhedra and directed graphs, termed alignments. We demonstrate the polyhedra of any given alignment can always be realized as compact, orientable surfaces. For any  n ∈ N, the class of quasigroups having associated surfaces with sum genus 6 n is closed under subobjects and homomorphic images. Further, we demonstrate semisymmetric quasigroup homomorphisms may be translated into branched covers between their respective surfaces.