David Blight's Race and Reunion touches on the subject of remembrances for the Civil War dead quite extensively. If I understand your question correctly, you are asking if Confederates mourned Union dead and vice-versa. Essentially the answer is no at first, and then yes later. Immediately after the war, there was quite a bit of acrimony (as you might imagine), and separate veterans' and other memorial organizations were formed. As Blight details, however, gradually white Americans became more interested in healing wounds than they were in belaboring the older grudges. In other words, white supremacy won out over sectional differences once everyone started living in a slavery-free America and experiencing the renewed prosperity brought by an economy with a heavily industrialized North and a plantation South that used sharecroppers rather than outright slaves. In the later years of the nineteenth century, after the official end of military Reconstruction, there started to be joint celebrations of the valor of soldiers that omitted the negative reasons that Southerners fought in favor of the values of the Lost Cause (love of the homeland, nobility, states' rights, etc).