This class of cell seems to care about orientation but not much else. These cells act like they are somehow or-ing together a bunch of simple cells, all sharing the same orientation, but differing in spatial position, phase, and symmetry. In contrast to simple cells, their response is not linear in the size of the preferred stimulus. They seem to respond equally well to a stimulus with the preferred orientation no matter what its size of polarity, and not matter where the stimulus is located in the receptive field.
In our program, we modeled a complex cell by summing together the outputs from two odd simple cells at the same location but with opposite polarity. You can see how this results in a cell that responds optimally to this orientation.