The author introduces a generalization of intertwining operators for modules of vertex operator algebras and studies their properties, in particular duality, skew symmetry and adjoint intertwining operators. This generalization is motivated by studying relations among certain twisted modules of a colored vertex operator superalgebra associated with an integral lattice; the construction is given in the second half of the paper. The main result is a description of fusion rules for certain twisted modules, a generalization of results in [C. Dong and J. Lepowsky, Generalized vertex algebras and relative vertex operators, Prog. Math. 112, Birkhäuser (1993; Zbl 0803.17009)].