Can two signals be run in opposite directions on one coaxial cable?

I won't bother explaining my entire situation as it is hard to describe but my question is, can I use the same coaxial cable that is bringing cable signal into one room of the house to go into a TiVo (actually it will be split going into two separate TiVos of different family members - we don't want to combine), then bring it back out of that TiVo and go through the same coax in the opposite direction (which it needs to do, to get downstairs)? I don't want to put the TiVo downstairs because it is a Bolt which is noisy, and the only cable jack is by the bed. The coax wiring is behind a wall. I also have ethernet at the same location of the TiVo which could reach downstairs, but it is by powerline adapter, and don't know if this would work to send TiVo signal downstairs. Complicated! Can signals run in opposite directions on the same coax?