Inspired by this question: Sexual reproduction without biological sex
And, to a lesser extent, this question: Would a society of simultaneous hermaphrodites have gender roles?
I'm not particularly well-versed in biology at all, let alone the wide variance of sexual characteristics that exists in the world, so I apologize if I've misunderstood/misused any terminology here. (Or if my questions are somehow just beyond stupid in a way that I failed to realize.)
For my purposes, I'm imagining a (preferably as close to human as possible) species that has exactly one biological sex. Every individual of this biological sex has all the necessary anatomical requirements to be the father or mother in a reproductive exchange. In case it's somehow relevant, their anatomy is also arranged in such a way that one couple can be simultaneously participating in two exchanges. This means that both members of one couple could be simultaneously pregnant with a child fathered by the other member of the couple.
So the main question here: If you have two children of the same two parents, but with swapped parental roles in their conception (the father of one is the mother of the other and vice versa), is there any inherently meaningful difference between the two siblings beyond two siblings who have the same parents as each other, both of whom are in the same parental roles?
I realize that a society comprised entirely of such a species could easily create social constructs such as children taking the surname of their biological mother or labeling such cross-siblings as something indicative of a relationship closer than a half-sibling but further than "actual" siblings. I'm less concerned with purely social constructs that might arise than I am with actual, meaningful differences that would affect them in the future more than "actual siblings" would be expected to experience.
If a social construct were to arise from such differences (as I expect they would), then I'm definitely interested in those sorts of downstream effects. In fact, the main reason I'm asking the question is so that I can eventually come up with exactly those kinds of downstream effects. But I'd like to set the foundation of my social constructs on something more solid than just other social constructs.
For instance, as AlexP brought up in comments, the intra-uterine environment can be very important because of genomic imprinting and/or metabolic imprinting. Aside from potential disorders that might happen at the statistical fringe, what noticeably-common differences might we see between cross-siblings? Since both parents could nurse the child, would some differences be mitigated by having a child's father nurse them instead of the mother? Would differences in mitochondrial DNA cause significant differences? Are any noticeably-visible or significant characteristics only heritable from one parent? These are all sub-questions of which I'm unsure of their relevance to the larger discussion.
I wish I could be more specific in what I'm looking for, but I'm struggling to find the words to describe it at the moment. Hopefully, I can think of a better way to word all of this fairly soon and edit my question accordingly.