Without more context I can't think of much to say about odd and even functions.
The distinction seems to be largely one of how you can treat them in various mathematical tricks - whether you can integrate from -infinity to + infinity in one go, or whether you need to do it in two halves, for eg.
I really don't know if there is some deep general significance to whether a function is odd or even. The issue just crops up sometimes when doing the maths.
I'd probably get in trouble with people like LvW for saying it, but I don't think maths is reality. It's all just models and often very imperfect models. The maths model has to take account of the real world, but the real world does not necessarily have to take account of the maths. Sometimes maths reveals some insight into the world that we did not have before, but sometimes things that crop up are just artefacts or quirks or imperfections in the models. In maths we do some pretty incredible things - pulses that last for zero time, have finite size but stil have power; signals that exist from - inf to + inf ; signals that instantaneously change from one value to another. None of these seem physically plausible, but doing maths with them can give useful results. It's hardly surprising that they sometimes give impossible results or require us to do other weird things to make them work.
I just don't worry about it. If it reliably gives us the useful results we need, use it. If it seems to suggest something weird, well, think about it a bit in case it is important, but if it's just weird, ignore it.
