Yes, you can install an omni-directional antenna upside down without affecting performance. An omni-directional antenna can pull in signals from any direction.
The signal pattern for an omni-directional antenna will work well as long as the antenna is vertical, that is, pointing straight up or straight down. However, they lose much of their effectiveness if they are mounted sideways or at an angle.
This answer applies only to omni-directional antennas. A directional antenna must be aimed in a specific direction to be effective.