  In this paper we introduce a particular class of Heffter arrays, called
globally simple Heffter arrays, whose existence gives at once orthogonal cyclic
cycle decompositions of the complete graph and of the cocktail party graph. In
particular we provide explicit constructions of such decompositions for cycles
of length $k\leq 10$. Furthermore, starting from our Heffter arrays we also
obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.
