I dont have much experience with animal spirit symbols in the Context of Reiki, however I do in my practice of shamanic healing. The wonderful thing about symbols( including animal symbolism) is that they mean many different things to many different people. Trust your intuition, In MY own practice when I see bugs in a specific area of clients body/Energy field it indicates some sort of Illness and unbalance but thats just ME!!! ( Because I HATE BUGS! so it makes sense to me ) when I have doubt about the meaning of an intuitive impression or symbol I meditate or do some journey work and ask my guides to help me make sense of it. So if I was in your situation I would ask my Reiki Guides for help on understanding the message that the spider symbol was trying to convey and keep alert for their response.
From Beyond the Rainbow
RD
RMT.