I'm not sure what you mean by symmetrical. From reading your post, it looks like another way to say balanced. I'll assume thats the case.
As far as phono cartridges go, they are balanced by design. There may be some exceptions, but I'm not aware of any. The tone arm cables can be terminated balanced or single ended. If its terminated balanced, it should be true balanced, not just attaching an xlr connector to component or cable that is not balanced by design.
As far as the interconnects go, you are correct in that not all cables are balanced even if they are terminated with xlr connectors. If you have fully balanced equipment, you need to make sure your cables are fully balanced as well. The cable should have 3 separate conductors or 2 separate conductors and a drain wire. Anything less than that and you don't have a fully balanced cable.
As far as phono cartridges go, they are balanced by design. There may be some exceptions, but I'm not aware of any. The tone arm cables can be terminated balanced or single ended. If its terminated balanced, it should be true balanced, not just attaching an xlr connector to component or cable that is not balanced by design.
As far as the interconnects go, you are correct in that not all cables are balanced even if they are terminated with xlr connectors. If you have fully balanced equipment, you need to make sure your cables are fully balanced as well. The cable should have 3 separate conductors or 2 separate conductors and a drain wire. Anything less than that and you don't have a fully balanced cable.