In this manuscript, we employ interface enhanced computational homogenization to explore and detail on a number of unfamiliar characteristics that composites can exhibit at different length scales. Here, the interface between the constituents is general in the sense that both displacement and traction jumps across the interface are admissible. We carry out numerous computational investigations using the finite element method for a broad range of various material parameters. Our numerical results reveal that the effective response of a microstructure embedding general interfaces is intuitively unpredictable and highly complex. In particular, for certain ranges of material parameters the overall response shows insensitivity with respect to either microstructure size or stiffness-ratio between inclusion and matrix. This unique behavior is observed likewise for two- and three-dimensional unit-cells. Our findings provide a valuable guideline to design tunable composites utilizing interfaces.