It depends on antenna feeding/excitation
(usually ... In some cases it means geometry of the antenna too. You must know what is the point of asymmetricity).
For example: If you need to drive antenna by "+/-" signal (like dipole antenna), you have usually symetrical one because you must drive one side of antenna by "+" signal from the source and the second side by "-" signal.
(One thing - "+" and "-" source means that the "+" source is shifted from "-" source by 180 degrees - no DC !
)
Example: dipole antenna is symetrical (you need "+" and "-" source). But monopole antenna is non-symetrical. You drive it only by one source (the second side is "mirrored through the ground")
If you want to know what type of antenna you have, you must know 2 things.
1. what is your source? (are you feeding antenna by two signals that are shifted to each other? Or you use only one signal?)
2. Is there some sort of conversion before the antenna itself? (e.g. balun or 180 phase loop)
(I hope that it help you and my answer is not more confusing for you - read some articles, on wiki for example, there are basic principles you need to know and they are easy to understand)
EDIT: vfone is right, patch Antenna is not symmetrical (2nd post below)
(I mixed up ordinary planar with patch antenna). Moreover, he is right that "symetry is related o the current/phase distribution on the antenna" ... geometry is product of desired current distribution (it is much better explanation than mine
).