Is it possible to have constipated diarrhea?

Actually--yes: If the constipation is long term it builds up fecal material in the colon. This can subsequentlly partially liquefy and lead to diarrhea. I had a patient whose left leg was paralyzed due to a massive build up of stool(it took about 10 enemas to clear it) that pressed on the motor nerve to her leg. The paralysis cleared up after the stool was removed.

Yes: This is called "overflow diarrhea" in which liquid stool that sits above a hard, impacted, constipated stool mass in the rectum, leaks around this mass and looks like diarrhea. However, it usually is frequent and small volume.