In the center of a Gaussian beam, the field structure is close to that in a plane wave with the same polarization. So the field is not axially symmetric for a linearly polarized Gaussian beam.
Let me note that Gaussian beams are not precise solutions of the free Maxwell equations. For this reason, a few years ago, I derived some precise solutions of the free Maxwell equations that are close to Gaussian beams in some sense (http://arxiv.org/abs/physics/0405091 , eq.22). I considered circular polarization only, but it is not difficult to build solutions with linear polarization starting from the solutions with circular polarization.
EDIT (07/04/2013): I am trying here to answer lurscher's question in the comments. It depends on where you place the surface of the reflecting material. If you place it in the center of the waist of the Gaussian beam, where the structure of the field is close to that in a plane wave, the reflection coefficients will be very close to those for a plane wave, I believe. If the reflecting surface is far from the center of the waist, the structure of the field will be close to that in radiation from a point source (within some angle of divergence). Directions of propagation will span some solid angle, so it seems the only way to eliminate s-polarization completely is to use a surface orthogonal to the axis of the Gaussian beam, however, in this case, reflection will be relatively small. On the other hand, if the Gaussian beam is narrow, the share of s-polarized radiation can be made small with a different choice of the reflecting surface. I don't have time right now to give more details.