Asymmetric verses Symmetric just refers to the paths that data takes, round trip. With redundant design traffic flows may follow two or more paths. The packets travelling from A to B may follow a different path than packets travelling back from B to A. In most cases this is of no particular concern. In some cases a high degree of determinism may be desired.
Routing protocol metrics can be tuned to ensure that packets leaving a building block follow the same path as packets returning to the building block. For example: Packets flow from station A.1 on VLAN A through its default gateway which is Layer 3 switch X. On Layer 3 switch X the routing metric on interface VLAN A is adjusted to make this path more favorable than the alternate return path through switch Y. If OSPF is the routing protocol, the interface cost metric is adjusted. If EIGRP is the routing protocol, the interface delay metric is adjusted. And so forth.
