Siblings (and other family members) often give birth order a lot of weight. Our twins were born by CS, and DH & I feel that it was arbitrary which one the obstetrician happened to grab first.
So for several years, we decided not to mention to anyone (even their grandparents) which was born first. It sounds a bit eccentric in retrospect, but we didn't want other people's birth order bias to affect their early development.
When they were of school age, the twins asked us who had been born first, so we told them, and let them make the decision about who to tell and when.