Symmetric lenses are as they sound; they have about the same distribution of positive and negative lenses through the lens and on either side of the aperture. This symmetry acts to place the principal planes near the center of the lens and near the actual aperture.
This type of design is used in quite a few photographic lenses, especiually those of a medium focal length. Those lenses that have a long enough focal length to not need to be retrofocus but not so long that they need to be made more compact with a telephoto design.
Other lenses (those that don't need to retain infinity focus or not even made to work on an SLR) will also commonly be symmetric such as enlarging lens (sometimes used on a bellows for macro work). Bellows lenses are also commonly relatively symmetric in design as they don't need to focus at inifinity.