Symmetric and asymmetric preferences
It is just a strict preference relation, an asymmetric (generic) relation $\succ$ is such that $a \succ b \implies \lnot b \succ a$. 
With respect to symmetric preferences there are not really  pure symmetric ones in consumer theory at least, the only symmetric relation is total indifference ~ over the whole choice set. Clearly a ~b implies b ~a. A typical $\succeq$ preference relation is the union of an asymmetric and symmetric part.