Rugate filters made of anisotropically nanostructured birefringent silicon have been fabricated and studied by polarization-resolved transmission measurements. Electrochemical etching of a (110) oriented Si wafer results in porous silicon layers which exhibit a strong in-plane birefringence. We demonstrate that a sinusoidal refractive index variation of birefringent porous silicon combined with index-matching layers and apodization results in a dichroic rugate filter having a stop-band dependent on the polarization direction of the incident light without higher-order harmonics and sidelobes. We also demonstrate that the combination of different dichroic rugate filters allow us to realize filters with more complex properties in a single preparation step.