First of all, you need to specify which system of ethics you want to base your judgement of incest on. People tend to think that “right” and “wrong” are absolutes (even when removed from religious considerations), when in fact “right” and “wrong” can only be specified with regards to specific systems of ethics, of which there are many.

Second, you need to define incest more precisely: In some cultures, incest includes relations between cousins, and between relatives by marriage, etc, while others consider only relations between direct blood relatives to be incest. Henceforth, I will consider only relations between siblings and parents to be incest.

In your question, you add the additional twist of seeking reasons “beyond science and religion”. This would rule out any consequentialist ethics like utilitarianism, since you are looking for something other than the negative consequences of incest, typically based on genetics and medical science. It would also rule out virtue ethics and some deontological ethics based on religious considerations.

This would leave us with non-religious deontological ethical systems as our only system for figuring out whether incest is wrong or not. I can only think of two such approaches:

Kant’s categorical imperative: An action is morally acceptable if it can be universalized, meaning that it should be allowed in every situation, not just the current one. So can incest be universalized? The answer is most likely “No”, and incest is wrong. There are far too many situations where allowing incest would disrupt family structures and relationships. In particular sexual partners can break up whenever the relationship no longer satisfies them, whereas the parent-child and sibling relationships are permanent ones that underly the very way our society is structured. Can you imagine a family holiday dinner where the a brother and sister had just broken up after a months long relationship? A man may divorce his wife to be with another woman, but what would happen to our social structure if men started divorcing their wives to be with their daughters?
Natural rights: Alternatively, one could argue from a point of view of natural freedom. As long as you are not harming others, you can do what you please. In this case, one has to admit that there is nothing wrong with incest per se. Presumably, those engaging in incestuous relationships would have to make sure that this never leads to procreation, since then they would be harming the eventual children by giving them a higher chance of genetics defects, but otherwise they are free, as consenting adults, to engage in whatever relationship they want to. Indeed, there are countries where incest between consenting adults is legal.