I have ancestors who fought on both sides, Virginia and New York, but those fellows were not related at the time. Nineteen confirmed members of my family fought for Virginia on my father's side and two confirmed members were New York Cav troopers on my mother's side.
I do not believe there were any more from my mom's side who might have fought in the war as the majority of the family at the time were in the Canadian Army, but there is a possibility.
On my dad's side it is quite possible more members of the family fought in the war and there are stories of a number of cousins who fought for the Union, but I have not been able to track them down and confirm it. It does make sense as the family was one of the founding families of what would later become Pendleton County WV and Highland County Virginia. At some point around 1840, the family started to migrate north west and ended up in Ohio, but we have not been able to get a good fix on the dates.
From my research doing our family tree, the family was very large from the late 18th- early 19th century, with an average of nine children per couple. It also appears as though 80% of the children born on my dad's side of the family were male, so there is an excellent chance we had close family members on opposite ends of muzzles.