symbols in the cave
The swirls are horses (or nature). The phi symbol is made of people (who evolved in Essos, originally).
Religion is obviously real in GoT, so I reckon it could come down to two different systems of belief. Ice refers to the Old Gods and Fire refers to the Lord of Light (I don't think we've seen any magic from the New Gods).
Maybe the White Walkers have overridden their programming like in every killer robot film, ever. They were originally supposed to only track the traces of the Lord of Light found in the first men. But eventually decided that all magic needed to die out... hence their war widening to both the Children of the Forest and human beings.
Obviously, my theory isn't properly fleshed out. But the more I think about it, the more I think 'Ice' and 'Fire' (as referenced in the title of the series) are tangents from the magic we've seen, rather than a reference to any characters.