Perpendicular-incidence ellipsometry (PIE) is considerably simpler, both analytically and in instrumentation, than oblique-incidence ellipsometry for the characterization of anisotropic surfaces and thin films. Here we describe how perpendicular-incidence null ellipsometry (PINE) can be used to completely determine the normalized reflection matrix, R = (rHyrvy, of a surface with arbitrary anisotropy. To separate the off-diagonal elements of the reflection matrix, a non-reciprocal optical element, e.g., a Faraday rotator, must be used. Instrumental data consist of azimuth angles of a linear polarizer and a linear (quarter-wave) retarder that produce four nulls, two nulls being acquired in the presence of a specific Faraday rotation, the other two in its absence. This extension makes possible the characterization by PIE of new optical surface anisotropies heretofore not considered, such as natural or induced surface optical activity and circular dichroism, or Kerr magneto-optic effects.