I shifted Duke Dawg's post around, added Tennis (shouldn't have), and added records (JMU win - ODU win - Tie). Added records for last 10 games and it looks like JMU is mostly moving in the right direction on the sports that ODU is leading and extending the advantage in sports JMU is leading.
ODU has a sizeable lead:
Mens Hoops (28-58). Last 10 is 4-6 (back to 2011)
Women's Hoops (22-51). Last 10 is 8-2 (back to 2009)
Field Hockey (7-19). Last 10 is 7-3 (back to 2011)
W Tennis (16-29) including the 1-20 since 1998 OUCH
M Tennis (2-32)
JMU has a sizeable lead:
Women's Soccer (16-6-1). Last 10 is 7-3 (back to 2005)
Lacrosse (36-8). Last 10 is 9-1 (back to 2004)
Women's Swimming and Diving (8-1)
Close / not a lot of action:
Men's Soccer (19-13-8). Last 10 is 3-4-3 (back to 2004)
Womens VBall 2-0
Football is 0-2
Baseball is 51-59 and 5-5 in last 10
Sports that didn't have historical records:
Men's Golf
Women's Golf
JMU sports without a ODU counter: Women's Track and Field (ODU seems to have an official club team -
https://odutrack.com/), Women's Cross Country (ODU dropped 2002)
ODU sports without a JMU counter: Men's and Women's Sailing, Women's Rowing, Men's Swimming (JMU dropped in 2006)
EDIT: Added more sports to answer Mturn017's question and updated men's soccer since ODU is trying to rewrite history.