The extraordinary light transmission through double-hole and elliptical nanohole arrays in a thin gold film is investigated for different orientations of the holes relative to the lattice. Even though these bases have similar symmetry characteristics, the polarization follows the orientation of the basis for the ellipse but remains fixed along a lattice vector for the double holes. Furthermore, the maximum transmitted intensity for linearly polarized light is constant for the ellipse, but decreases for the double holes as they are rotated away from being aligned with the lattice. Finite-difference time-domain simulations agree well with the experimental findings. These experiments show how the basis determines both the coupling into the surface plasmon waves and the evanescent transmission through the nanoholes. Both of these effects need to be considered when designing nanophotonic devices using the extraordinary transmission phenomenon.